Physics

Chapter 17 Waves

Imron Rosyadi

Chapter 16: Waves

Ocean Wave

Welcome to Chapter 16: Waves! Today, we’ll explore the fundamental concepts of waves, from how they travel to how they interact with each other and their surroundings. Waves are everywhere in our world, from the ocean waves you see to the sound waves you hear, and even the light waves that allow you to see.

16.1 Traveling Waves: Introduction

Waves are disturbances that propagate through a medium or space.
They transfer energy and momentum without transferring mass.
Think of a ripple in a pond: the water itself doesn’t travel across the pond, but the disturbance does.

To understand waves, let’s start with a simple definition. A wave is essentially a disturbance that moves from one place to another. The key idea here is that the wave itself moves, carrying energy and momentum, but the material through which it travels, the medium, does not move with the wave. Instead, the particles of the medium oscillate around their equilibrium positions.

16.1 Traveling Waves: Types of Waves

There are three basic types of waves:

1. Mechanical Waves:
   • Require a medium to propagate.
   • Governed by Newton’s laws.
   • Examples: Water waves, sound waves, seismic waves.
   • Transfer energy and momentum, not mass.
2. Electromagnetic Waves:
   • Do not require a medium (can travel through a vacuum).
   • Associated with oscillating electric and magnetic fields.
   • Examples: Light, radio waves, X-rays.
   • Travel at the speed of light (\(c = 2.99792458 \times 10^8 \text{ m/s}\)).

3. Matter Waves:
   • Central to quantum mechanics.
   • Associated with fundamental particles (protons, electrons, etc.).
   • First proposed by Louis de Broglie in 1924.
   • Discussed in quantum mechanics.

Let’s classify waves into three main categories. First, mechanical waves. These are the most intuitive for us as they require a physical medium to travel. Think about sound waves needing air or water waves needing water. The particles of the medium oscillate to transfer the disturbance. Next, we have electromagnetic waves. These are fascinating because they don’t need a medium; they can travel through the vacuum of space. Light is the most common example. They consist of oscillating electric and magnetic fields. Finally, there are matter waves, which are more abstract and come from quantum mechanics. These waves describe the wave-like properties of particles like electrons. We won’t go into detail about matter waves today, but it’s important to know they exist.

16.1 Traveling Waves: Characteristics of Mechanical Waves

Mechanical waves, particularly simple harmonic waves, can be described by several key characteristics:

• Wavelength (\(\lambda\)): The distance between two consecutive identical points on a wave (e.g., crest to crest). Measured in meters (m).
• Amplitude (\(A\)): The maximum displacement of the medium from its equilibrium position. Measured in meters (m).
• Period (\(T\)): The time it takes for one complete oscillation of a point in the medium, or for one wavelength to pass a point. Measured in seconds (s).
• Frequency (\(f\)): The number of oscillations per unit time. It’s the inverse of the period (\(f = 1/T\)). Measured in Hertz (Hz), where \(1 \text{ Hz} = 1 \text{ s}^{-1}\).
• Wave Speed (\(v\)): The speed at which the disturbance propagates through the medium.

\[v = \frac{\lambda}{T} = \lambda f\]

Now let’s delve into the specific characteristics we use to describe mechanical waves, especially simple harmonic waves, which repeat themselves. Imagine a wave on a string or a water wave. The wavelength, denoted by lambda (\(\lambda\)), is the spatial extent of one full cycle of the wave. It’s like measuring the distance from one wave crest to the next. The amplitude (\(A\)) tells us how “tall” the wave is, or the maximum displacement of the medium from its resting position. The period (\(T\)) is about time – how long it takes for one complete wave cycle to pass a point. The frequency (\(f\)) is the inverse of the period and tells us how many cycles pass per second. And finally, wave speed (\(v\)) is how fast the wave itself travels. This is a fundamental relationship connecting wavelength, frequency, and wave speed.

16.1 Traveling Waves: Transverse and Longitudinal Waves

Transverse Waves:

• The disturbance of the medium is perpendicular to the direction of wave propagation.
• Examples: Waves on a string, surface water waves.

Caption: (a) Transverse wave

Longitudinal Waves (Compressional Waves):

• The disturbance of the medium is parallel to the direction of wave propagation.
• Examples: Sound waves in air, P-waves in seismic activity.

Caption: (b) Longitudinal wave

Mechanical waves can also be categorized by how the medium’s particles move relative to the wave’s direction of travel. In transverse waves, the particles oscillate perpendicular to the direction the wave is moving. Imagine shaking a rope up and down to create a wave that travels horizontally. The rope segments move up and down, while the wave moves sideways. In longitudinal waves, the particles oscillate parallel to the direction of wave propagation. Think of a Slinky being pushed and pulled. The compressions and expansions travel along the Slinky, and the coils themselves move back and forth in the same direction. Sound waves are a great example of longitudinal waves.

16.2 Mathematics of Waves: Wave Function

A sinusoidal wave traveling in the positive \(x\)-direction can be modeled by the wave function:

\[y(x,t) = A \sin(kx - \omega t + \phi)\]

Where:

• \(A\): Amplitude (maximum displacement)
• \(k\): Wave number (\(k = \frac{2\pi}{\lambda}\))
• \(\omega\): Angular frequency (\(\omega = \frac{2\pi}{T} = 2\pi f\))
• \(\phi\): Phase shift (initial phase of the wave)
• \(kx - \omega t + \phi\): Phase of the wave

For waves moving in the negative \(x\)-direction, the equation is \(y(x,t) = A \sin(kx + \omega t + \phi)\).

The wave speed is related to \(\omega\) and \(k\):

\[v = \frac{\omega}{k}\]

To describe waves mathematically, we use a concept called a wave function, \(y(x,t)\). This function tells us the displacement of the medium (y) at any given position (x) and time (t). For a simple harmonic (sinusoidal) wave moving in the positive x-direction, the wave function often takes the form \(A \sin(kx - \omega t + \phi)\). Let’s break down these terms: \(A\) is the amplitude, as we’ve discussed. \(k\) is the wave number, which relates to the wavelength. A larger \(k\) means a shorter wavelength. \(\omega\) is the angular frequency, related to the period and frequency. A larger \(\omega\) means a higher frequency. \(\phi\) is the phase shift, which accounts for the wave’s starting position at \(t=0, x=0\). The entire argument inside the sine function, \(kx - \omega t + \phi\), is called the phase. If the wave is moving in the negative x-direction, the sign in front of \(\omega t\) changes to a plus. Crucially, the wave speed (\(v\)) is also linked to the angular frequency and wave number by the simple relation \(v = \omega / k\).

16.2 Mathematics of Waves: Medium Velocity and Acceleration

For a transverse wave described by \(y(x,t) = A \sin(kx - \omega t + \phi)\):

Velocity of the medium particles (\(v_y\)):
(Found by taking the partial derivative of \(y\) with respect to \(t\))

\[v_y(x,t) = \frac{\partial y}{\partial t} = -A\omega \cos(kx - \omega t + \phi)\]

The maximum velocity of the medium is \(|v_{y,max}| = A\omega\).

Acceleration of the medium particles (\(a_y\)):
(Found by taking the partial derivative of \(v_y\) with respect to \(t\))

\[a_y(x,t) = \frac{\partial v_y}{\partial t} = -A\omega^2 \sin(kx - \omega t + \phi)\]

The maximum acceleration of the medium is \(|a_{y,max}| = A\omega^2\).

Note

The wave speed (\(v\)) is constant, but the velocity and acceleration of the medium particles are not. They oscillate in simple harmonic motion.

It’s important to distinguish between the wave speed and the speed of the particles in the medium. The wave propagates at a constant speed, but the individual particles of the medium oscillate around their equilibrium positions. For a transverse wave, these oscillations are perpendicular to the wave’s direction of travel. We can calculate the velocity of these oscillating particles by taking the partial derivative of the wave function with respect to time. This gives us \(v_y(x,t) = -A\omega \cos(kx - \omega t + \phi)\). Notice that the maximum speed of a particle is \(A\omega\), which should be familiar from simple harmonic motion. Similarly, the acceleration of the medium particles can be found by taking another partial derivative with respect to time, giving \(a_y(x,t) = -A\omega^2 \sin(kx - \omega t + \phi)\). The maximum acceleration is \(A\omega^2\). So, while the wave rushes along at its constant speed, the tiny bits of the medium are busy doing their own simple harmonic dance.

16.2 Mathematics of Waves: The Linear Wave Equation

The relationship between the second partial derivatives of the wave function with respect to position and time yields the linear wave equation:

\[\frac{\partial^2 y(x,t)}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y(x,t)}{\partial t^2}\]

Where:

• \(\frac{\partial^2 y(x,t)}{\partial x^2}\) is the curvature of the wave.
• \(\frac{\partial^2 y(x,t)}{\partial t^2}\) is the acceleration of the medium.

Important

Any wave function \(y(x,t) = f(x \mp vt)\) that satisfies this equation is a linear wave function.
The linear wave equation is fundamental for describing waves in various physical systems (e.g., waves on a string, sound waves, electromagnetic waves).

One of the most profound mathematical descriptions in physics related to waves is the linear wave equation. This equation connects how the curvature of a wave (how its shape changes with position) is related to how the medium accelerates (how its speed changes with time). Specifically, it states that the second partial derivative of the wave function with respect to position is proportional to the second partial derivative with respect to time, with the constant of proportionality being \(1/v^2\), where \(v\) is the wave speed. Any function that satisfies this equation can represent a wave. This equation is incredibly powerful because it applies to a wide variety of waves, from mechanical waves like those on a string to sound waves and even electromagnetic waves. A crucial property of linear wave functions is that if two waves are solutions to this equation, their sum is also a solution. This leads us directly to the principle of superposition.

16.2 Mathematics of Waves: Principle of Superposition

If two or more linear waves combine at the same point in a medium, the resulting displacement of the medium at that point is the algebraic sum of the displacements due to the individual waves.

If \(y_1(x,t)\) and \(y_2(x,t)\) are solutions to the linear wave equation, then \(y_R(x,t) = y_1(x,t) + y_2(x,t)\) is also a solution.

This property is known as the principle of superposition.
It allows waves to interfere constructively or destructively.

The principle of superposition is a cornerstone of wave physics. It states that when two or more waves overlap in the same medium, the total disturbance at any point is simply the algebraic sum of the individual disturbances from each wave. This is why we can combine wave functions by simply adding them. This principle is key to understanding phenomena like interference, where waves either reinforce each other (constructive interference) or cancel each other out (destructive interference). We’ll explore interference in more detail later.

16.3 Wave Speed on a Stretched String

The speed of a wave on a stretched string depends on two main factors:

1. Tension (\(F_T\)): The force pulling on the string.

2. Linear Mass Density (\(\mu\)): The mass per unit length of the string.

   \[\mu = \frac{\text{mass of string}}{\text{length of string}} = \frac{m}{L}\]

The wave speed (\(v\)) on a string under tension is given by:

\[v = \sqrt{\frac{F_T}{\mu}}\]

Tip

In general, the speed of a wave in a medium is determined by the elastic property (how easily the medium restores itself) and the inertial property (how much mass resists motion) of the medium:

\[v = \sqrt{\frac{\text{elastic property}}{\text{inertial property}}}\]

Now let’s apply our understanding of wave speed to a specific and very practical scenario: waves on a stretched string. This is crucial for understanding musical instruments like guitars and violins. The speed of a wave on a string isn’t just arbitrary; it’s determined by the physical properties of the string itself. The first factor is the tension (\(F_T\)) in the string. The tighter the string, the faster the waves will travel. The second factor is the linear mass density (\(\mu\)), which is the string’s mass per unit length. A thicker, heavier string (higher linear density) will result in slower wave speeds. The formula linking these is \(v = \sqrt{F_T / \mu}\). This is a specific example of a more general principle: wave speed in any medium depends on the medium’s elastic property (how it “wants” to return to its original state) and its inertial property (how much it resists being moved). For a string, tension provides the restoring force (elastic property), and linear density provides the inertia.

16.4 Energy and Power of a Wave

Waves carry energy. The amount of energy and the rate of energy transfer (power) are related to the wave’s characteristics.

Time-averaged power (\(P_{ave}\)): The average rate of energy transfer for a sinusoidal mechanical wave.

\[P_{ave} = \frac{1}{2} \mu A^2 \omega^2 v\]

Where:

• \(\mu\): Linear mass density
• \(A\): Amplitude
• \(\omega\): Angular frequency
• \(v\): Wave speed

Note

The power of a mechanical wave is proportional to:

• The square of the amplitude (\(A^2\))
• The square of the angular frequency (\(\omega^2\)) (or frequency \(f^2\))
• The wave speed (\(v\))

One of the most important aspects of waves is that they transport energy. We see this in everything from destructive earthquakes to the warmth of sunlight. For a mechanical wave, the energy carried is related to its amplitude and frequency. Higher amplitude waves carry more energy, and higher frequency waves deliver energy more rapidly. The time-averaged power, which is the average rate at which energy is transferred by a sinusoidal mechanical wave, is given by the formula \(P_{ave} = \frac{1}{2} \mu A^2 \omega^2 v\). This formula shows a critical relationship: the power is proportional to the square of the amplitude and the square of the angular frequency. This means doubling the amplitude or doubling the frequency increases the power by a factor of four! The power is also directly proportional to the wave speed.

16.4 Energy and Power of a Wave: Intensity

Intensity (\(I\)): Power per unit area.

\[I = \frac{P}{A_{rea}}\]

Measured in watts per square meter (\(\text{W/m}^2\)).

For a spherical wave (e.g., sound from a speaker), the area \(A_{rea} = 4\pi r^2\).

\[I = \frac{P}{4\pi r^2}\]

Tip

As a spherical wave moves away from its source, its intensity decreases with the square of the distance from the source (\(I \propto \frac{1}{r^2}\)), assuming no energy dissipation.

Beyond just power, we often talk about intensity, which describes how concentrated the wave’s power is. Intensity is defined as power per unit area. Its units are watts per square meter. This concept is especially useful for waves that spread out, like sound from a speaker or light from a bulb. For a spherical wave, the energy spreads over the surface of an expanding sphere, so the area is \(4\pi r^2\). This means that as you move further away from a source (increase \(r\)), the intensity of the wave decreases rapidly, proportional to \(1/r^2\). This is why a sound seems quieter when you’re further away from the source.

16.5 Interference of Waves: Reflection and Transmission

When a wave encounters a boundary of the medium, it can be reflected and/or transmitted.

Fixed Boundary Condition:

• Medium is fixed in place.
• Reflected wave is 180° (\(\pi\) rad) out of phase with the incident wave (inverted).

Free Boundary Condition:

• Medium is free to move.
• Reflected wave is in phase with the incident wave.

When a wave moves from one medium to another (e.g., different linear densities):

• Both reflected and transmitted waves can occur.
• Phase relationships depend on whether the wave moves from a less dense to a more dense medium, or vice-versa.

What happens when a wave hits a wall or changes the medium it’s traveling through? This leads us to the concepts of reflection and transmission. When a wave hits a boundary, part of it bounces back (reflection) and part of it can pass through (transmission). The nature of the reflected wave depends on the boundary condition. If the boundary is fixed (like a string tied to a wall), the reflected wave is inverted, meaning it’s \(180^\circ\) out of phase with the incident wave. A crest reflects as a trough. If the boundary is free (like a string attached to a light ring on a frictionless pole), the reflected wave is in phase. A crest reflects as a crest. When a wave encounters a change in medium, like a thin string connected to a thick string, both reflection and transmission occur. The phase of the reflected wave again depends on whether it’s going from a “lighter” to “heavier” medium or vice-versa.

16.5 Interference of Waves: Constructive and Destructive Interference

When two or more waves overlap, their displacements add algebraically (Principle of Superposition). This phenomenon is called interference.

Constructive Interference:

• Waves arrive in phase (crests align with crests, troughs with troughs).
• Resultant wave has an amplitude greater than the individual waves.
• Maximum constructive interference occurs when waves are exactly in phase (\(0^\circ\) phase difference), leading to an amplitude of \(2A\) for two identical waves.

Destructive Interference:

• Waves arrive out of phase (crests align with troughs).
• Resultant wave has an amplitude smaller than the individual waves.
• Maximum destructive interference occurs when waves are exactly \(180^\circ\) (\(\pi\) rad) out of phase, leading to zero amplitude for two identical waves.

Interference is a fundamental wave phenomenon where two or more waves overlap and combine their effects. It’s a direct consequence of the principle of superposition. There are two main types of interference. Constructive interference happens when waves arrive at a point in phase, meaning their crests line up with crests and troughs with troughs. The displacements add up, resulting in a larger amplitude. If two identical waves are perfectly in phase, the resulting wave has double the amplitude. Destructive interference occurs when waves arrive out of phase, like a crest meeting a trough. In this case, the displacements subtract, leading to a smaller amplitude. If two identical waves are perfectly out of phase (\(180^\circ\) phase difference), they completely cancel each other out, resulting in zero amplitude. Most real-world interference involves a mix of constructive and destructive effects, leading to complex patterns, like the varying loudness of sound in a room.

16.5 Interference of Waves: Superposition of Sinusoidal Waves

Two identical sinusoidal waves differing by a phase shift \(\phi\):

\(y_1(x,t) = A \sin(kx - \omega t + \phi)\)
\(y_2(x,t) = A \sin(kx - \omega t)\)

The resultant wave \(y_R(x,t) = y_1(x,t) + y_2(x,t)\) is:

\[y_R(x,t) = [2A \cos(\frac{\phi}{2})] \sin(kx - \omega t + \frac{\phi}{2})\]

The amplitude of the resultant wave is \(A_R = 2A \cos(\frac{\phi}{2})\).

• If \(\phi = 0\) (in phase), \(A_R = 2A\) (constructive interference).
• If \(\phi = \pi\) (out of phase), \(A_R = 0\) (destructive interference).
• If \(\phi = \pi/2\), \(A_R = \sqrt{2}A\).

Let’s apply the principle of superposition to two specific sinusoidal waves that are identical in amplitude, wavelength, and frequency, but differ only by a phase shift \(\phi\). When we add these two waves together using a trigonometric identity, we get a new resultant wave. The key outcome is that this resultant wave still has the same wave number and angular frequency, but its amplitude is now \(A_R = 2A \cos(\phi/2)\), and its phase is shifted by \(\phi/2\). This formula beautifully illustrates constructive and destructive interference: If \(\phi = 0\) (meaning the waves are in phase), \(\cos(0) = 1\), so \(A_R = 2A\), which is maximum constructive interference. If \(\phi = \pi\) (meaning the waves are \(180^\circ\) out of phase), \(\cos(\pi/2) = 0\), so \(A_R = 0\), which is complete destructive interference. Any other phase difference will result in an amplitude somewhere between 0 and \(2A\).

16.6 Standing Waves and Resonance

Standing Waves:

• Formed by the superposition of two identical traveling waves moving in opposite directions.
• Appear to vibrate in place, without propagating.
• Characterized by nodes (points of zero displacement) and antinodes (points of maximum displacement).

The wave function for a standing wave is:

\[y(x,t) = [2A \sin(kx)] \cos(\omega t)\]

• Nodes occur where \(\sin(kx) = 0\), i.e., \(x = n \frac{\lambda}{2}\) for \(n=0, 1, 2, \ldots\).
• Antinodes occur where \(\sin(kx) = \pm 1\), i.e., \(x = n \frac{\lambda}{4}\) for \(n=1, 3, 5, \ldots\).

Sometimes, waves don’t seem to travel; instead, they appear to vibrate in place. These are called standing waves. Standing waves are created when two identical traveling waves move in opposite directions and superimpose. Think of plucking a guitar string: waves travel down the string, reflect at the ends, and interfere to create a standing wave pattern. The mathematical form of a standing wave shows this: \(y(x,t) = [2A \sin(kx)] \cos(\omega t)\). Notice how the position-dependent term (\(\sin(kx)\)) is separated from the time-dependent term (\(\cos(\omega t)\)). This equation reveals key features: Nodes are points on the wave that always have zero displacement. They occur at fixed positions where \(\sin(kx)=0\). Antinodes are points where the displacement is maximum. They occur where \(\sin(kx)=\pm 1\). These nodes and antinodes are characteristic of standing waves and remain fixed in space, unlike the crests and troughs of traveling waves.

16.6 Standing Waves and Resonance: Normal Modes

For a string of length \(L\) with fixed ends (nodes at both ends):

The possible wavelengths for standing waves are:

\[\lambda_n = \frac{2L}{n}, \quad \text{for } n = 1, 2, 3, \ldots\]

The corresponding resonant frequencies (normal frequencies) are:

\[f_n = \frac{nv}{2L} = n f_1, \quad \text{for } n = 1, 2, 3, \ldots\]

Where \(f_1 = \frac{v}{2L}\) is the fundamental frequency (or first harmonic).

• \(n=1\): First harmonic (fundamental mode), \(\lambda_1 = 2L\)
• \(n=2\): Second harmonic (first overtone), \(\lambda_2 = L\)
• \(n=3\): Third harmonic (second overtone), \(\lambda_3 = \frac{2}{3}L\)

Standing waves are closely related to the concept of resonance. When a string or any system is driven at specific frequencies, it can resonate, producing large-amplitude standing waves. These specific patterns are called normal modes. For a string fixed at both ends (a common scenario for musical instruments), the boundary conditions require nodes at each end. This limits the possible wavelengths and frequencies that can form stable standing waves. The possible wavelengths are \(\lambda_n = 2L/n\), where ‘n’ is an integer representing the mode number. Correspondingly, the resonant frequencies are \(f_n = nv/(2L)\). The \(n=1\) mode is called the fundamental frequency or first harmonic, which has a wavelength of \(2L\). Higher values of ‘n’ correspond to overtones or higher harmonics. For example, \(n=2\) is the second harmonic (first overtone), \(n=3\) is the third harmonic (second overtone), and so on. These different modes produce different pitches in musical instruments.

Key Takeaways

• Waves transfer energy and momentum without transferring mass.
• Mechanical waves require a medium, while electromagnetic waves do not.
• Waves are characterized by amplitude (\(A\)), wavelength (\(\lambda\)), period (\(T\)), frequency (\(f\)), and wave speed (\(v\)).
• The relationship \(v = \lambda f\) is fundamental.
• Waves can be transverse (disturbance perpendicular to propagation) or longitudinal (disturbance parallel to propagation).
• The wave function \(y(x,t) = A \sin(kx \mp \omega t + \phi)\) describes sinusoidal waves.
• The linear wave equation \(\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}\) applies to all linear waves.
• The principle of superposition states that overlapping waves add algebraically.
• Interference can be constructive (amplitudes add) or destructive (amplitudes subtract).
• Standing waves are formed by two identical waves moving in opposite directions, creating fixed nodes and antinodes.
• Resonance occurs when a system is driven at its natural (normal mode) frequencies, leading to large-amplitude standing waves.

Let’s summarize the key points from this chapter. Remember that waves are primarily about transporting energy and momentum, not the medium itself. We distinguish between mechanical waves, which need a medium, and electromagnetic waves, which don’t. The five key descriptors of waves are amplitude, wavelength, period, frequency, and wave speed, all linked by \(v = \lambda f\). Know the difference between transverse and longitudinal waves based on particle motion relative to wave direction. The wave function provides a mathematical model for sinusoidal waves, and the linear wave equation is a universal descriptor for many types of waves. Superposition is crucial: waves add up when they overlap, leading to constructive or destructive interference. Finally, standing waves are special cases where waves appear stationary, characterized by nodes and antinodes, and resonance explains how specific frequencies lead to large-amplitude oscillations in systems like musical strings.

Key Equations

Equation	Description
\(v = \lambda f\)	Wave speed in terms of wavelength and frequency
\(k = \frac{2\pi}{\lambda}\)	Wave number
\(\omega = 2\pi f = \frac{2\pi}{T}\)	Angular frequency
\(y(x,t) = A \sin(kx \mp \omega t + \phi)\)	Sinusoidal wave function
\(v_y(x,t) = -A\omega \cos(kx \mp \omega t + \phi)\)	Velocity of medium particles
\(a_y(x,t) = -A\omega^2 \sin(kx \mp \omega t + \phi)\)	Acceleration of medium particles
\(\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}\)	Linear wave equation
\(v = \sqrt{\frac{F_T}{\mu}}\)	Wave speed on a string under tension
\(P_{ave} = \frac{1}{2} \mu A^2 \omega^2 v\)	Time-averaged power of a sinusoidal mechanical wave
\(I = \frac{P}{A_{rea}}\)	Intensity (Power per unit area)
\(y_R(x,t) = [2A \cos(\frac{\phi}{2})] \sin(kx - \omega t + \frac{\phi}{2})\)	Superposition of two phase-shifted sinusoidal waves
\(y(x,t) = [2A \sin(kx)] \cos(\omega t)\)	Standing wave function
\(\lambda_n = \frac{2L}{n}\)	Wavelengths of normal modes on a string (fixed ends)
\(f_n = \frac{nv}{2L} = n f_1\)	Frequencies of normal modes on a string (fixed ends)

This table summarizes the most important equations from Chapter 16. These are the formulas you’ll use to solve problems and understand quantitative relationships in wave phenomena. Make sure you know what each variable represents and when to apply each equation.

Key Terms

Term	Definition
Wave	A disturbance that propagates, transferring energy and momentum without transferring mass.
Mechanical Wave	A wave that requires a medium to propagate.
Electromagnetic Wave	A wave associated with oscillations in electric and magnetic fields; does not require a medium.
Amplitude (\(A\))	The maximum displacement of the medium from its equilibrium position.
Wavelength (\(\lambda\))	The distance between two consecutive identical points on a wave.
Period (\(T\))	The time it takes for one complete oscillation or for one wavelength to pass a point.
Frequency (\(f\))	The number of oscillations per unit time (\(f=1/T\)).
Wave Speed (\(v\))	The speed at which the wave disturbance propagates through the medium.
Transverse Wave	A wave in which the disturbance of the medium is perpendicular to the direction of propagation.
Longitudinal Wave	A wave in which the disturbance of the medium is parallel to the direction of propagation (compressional wave).
Wave Function	A mathematical model describing the position of particles of the medium for every position and time.
Wave Number (\(k\))	A measure of the spatial frequency of a wave, equal to \(2\pi/\lambda\).
Angular Frequency (\(\omega\))	A measure of the rate of oscillation in radians per second, equal to \(2\pi f\).
Linear Wave Equation	A fundamental differential equation describing the propagation of linear waves.
Principle of Superposition	States that when two or more linear waves combine, the resultant displacement is the algebraic sum of individual displacements.
Interference	The phenomenon where two or more waves overlap, resulting in constructive or destructive effects.
Constructive Interference	Occurs when waves combine to produce a larger amplitude.
Destructive Interference	Occurs when waves combine to produce a smaller (or zero) amplitude.
Standing Wave	A wave that appears to vibrate in place, formed by the superposition of two identical waves moving in opposite directions.
Node	A point on a standing wave where the displacement is always zero.
Antinode	A point on a standing wave where the displacement is maximum.
Resonance	A phenomenon where a system’s amplitude of oscillation greatly increases when driven at its natural frequency.
Fundamental Frequency	The lowest natural frequency of vibration for a system, corresponding to the first harmonic (\(n=1\)).
Overtone	Any resonant frequency above the fundamental frequency.
Harmonic	A natural frequency that is an integer multiple of the fundamental frequency.