The displacement of the medium in a sound wave is given by the equation y_i = A cos(ax + bt) where A

Q: Why does the wave $y_i = A \cos(ax + bt)$ travel in the negative $x$-direction?

In any wave function of the form $f(kx \pm \omega t)$, if the signs of the spatial term $kx$ and the temporal term $\omega t$ are the same (both positive or both negative), the wave propagates in the negative $x$-direction. If they have opposite signs, the wave propagates in the positive $x$-direction.

Q: How did we get the amplitude of the reflected wave as $0.8 A$?

The intensity of a wave is proportional to the square of its amplitude ($I \propto A^2$). Since the reflected intensity is $0.64$ times the incident intensity ($I_r = 0.64 I_i$), the reflected amplitude is $A_r = \sqrt{0.64} A = 0.8 A$.

Q: Why is there a minus sign in the reflected wave equation $y_r = -0.8 A \cos(ax - bt)$?

When a wave reflects from a rigid boundary (denser medium), it undergoes a phase change of $\pi$ radians. Mathematically, this adds $\pi$ to the phase: $\cos(ax - bt + \pi)$. Since $\cos(\theta + \pi) = -\cos(\theta)$, this introduces the negative sign.

Q: How do we find the maximum and minimum particle speeds?

We first find the particle velocity $v_p$ by taking the partial derivative of the resultant displacement $y$ with respect to time $t$. The maximum speed occurs when the trigonometric terms align constructively to give the maximum possible magnitude, while the minimum speed is zero when they cancel out or vanish.

Q: What is the physical significance of having both a standing wave and a travelling wave component in the resultant wave?

This indicates partial reflection. A pure standing wave only occurs when reflection is $100\%$ efficient ($A_r = A$). When reflection is partial ($A_r < A$), some energy is reflected (forming the standing wave component) while the remaining unreflected energy continues to propagate (forming the travelling wave component).

Question

The displacement of the medium in a sound wave is given by the equation $y_i = A \cos (ax + bt)$ where $A$, $a$ and $b$ are positive constants. The wave is reflected by an obstacle situated at $x = 0$. The intensity of the reflected wave is $0.64$ times that of the incident wave.
(a) What are the wavelength and frequency of incident wave?
(b) Write the equation for the reflected wave.
(c) In the resultant wave formed after reflection, find the maximum and minimum values of the particle speeds in the medium.
(d) Express the resultant wave as a superposition of a standing wave and a travelling wave. What are the positions of the antinodes of the standing wave? What is the direction of propagation of travelling wave?

Sigma Percentile · Accepted Answer

### Introduction: The Dance of Waves and Boundaries

When a wave travels through a medium, it carries energy and momentum.

But what happens when this wave encounters a boundary?

This is one of the most fascinating questions in wave mechanics.

Depending on the nature of the boundary, the wave can be fully transmitted, fully reflected, or partially both.

In this problem, we explore the partial reflection of a sound wave at a rigid obstacle situated at $x = 0$.

We will see how the superposition of the incident and reflected waves creates a complex wave pattern that is a mixture of a standing wave and a travelling wave.

Let's dive deep into the physics and mathematics behind this beautiful phenomenon.

### Part 1: Decoding the Incident Wave

We are given the equation of the incident sound wave:

$$y_i = A \cos(ax + bt)$$

Here, $A$ is the amplitude, and $a$ and $b$ are positive constants.

To understand this wave, let's compare it with the standard progressive wave equation:

$$y = A \cos(kx + \omega t)$$

By direct comparison, we find:

- The wave number $k = a$
- The angular frequency $\omega = b$

Since wavelength $\lambda$ is related to the wave number by $\lambda = \frac{2\pi}{k}$, we have:

$$\lambda = \frac{2\pi}{a}$$

Similarly, the frequency $f$ is related to the angular frequency by $f = \frac{\omega}{2\pi}$, giving:

$$f = \frac{b}{2\pi}$$

Notice the sign of the phase term $(ax + bt)$.

Since both the spatial term $ax$ and the temporal term $bt$ have the same sign, the wave is propagating in the **negative $x$-direction** (towards the left).

This means the wave is coming from the region $x > 0$ and travelling towards the obstacle at $x = 0$.

### Part 2: The Clash with the Wall – Boundary Conditions and Reflection

When the incident wave hits the rigid obstacle at $x = 0$, it undergoes reflection.

A rigid obstacle acts as a fixed boundary.

At a fixed boundary, the displacement of the medium particles must be zero at all times because the boundary cannot move.

To satisfy this boundary condition, the reflected wave must undergo a **phase change of $\pi$ radians** (or $180^\circ$ phase inversion).

This phase inversion physically means that a compression reflects as a compression, and a rarefaction reflects as a rarefaction in terms of pressure, but the displacement wave is inverted.

Furthermore, the reflected wave must travel in the opposite direction, which is the **positive $x$-direction**.

Therefore, the phase term for the reflected wave will be $(ax - bt)$.

We are also told that the intensity of the reflected wave is $0.64$ times that of the incident wave:

$$I_r = 0.64 I_i$$

Since the intensity of a wave is directly proportional to the square of its amplitude ($I \propto A^2$), we can write:

$$A_r^2 = 0.64 A^2 \implies A_r = 0.8 A$$

Now, we can write the complete equation for the reflected wave, incorporating the amplitude, the direction of propagation, and the phase shift of $\pi$:

$$y_r = 0.8 A \cos(ax - bt + \pi)$$

Using the trigonometric identity $\cos(	heta + \pi) = -\cos(	heta)$, this simplifies to:

$$y_r = -0.8 A \cos(ax - bt)$$

This is our reflected wave equation!

### Part 3: The Superposition Principle and Resultant Wave

According to the principle of superposition, when two waves overlap in a medium, the resultant displacement is the algebraic sum of the individual displacements:

$$y = y_i + y_r$$

Substituting our equations for $y_i$ and $y_r$:

$$y = A \cos(ax + bt) - 0.8 A \cos(ax - bt)$$

This equation represents the total wave field in the region $x > 0$.

It is a combination of a wave travelling to the left and a wave travelling to the right.

Because the amplitudes of the two waves are not equal ($A 
eq 0.8 A$), they do not form a pure standing wave.

Instead, they form a **partial standing wave**, which we will decompose shortly.

### Part 4: Particle Speeds – The Kinetic Reality

Let's find the velocity of the particles in the medium.

The particle velocity $v_p$ is the rate of change of displacement with respect to time:

$$v_p = \frac{\partial y}{\partial t}$$

Differentiating the resultant displacement equation with respect to $t$:

$$v_p = \frac{\partial}{\partial t} [A \cos(ax + bt) - 0.8 A \cos(ax - bt)]$$

$$v_p = -Ab \sin(ax + bt) - 0.8 Ab \sin(ax - bt)$$

To find the maximum and minimum particle speeds, we look at the magnitude of this velocity:

$$|v_p| = |Ab \sin(ax + bt) + 0.8 Ab \sin(ax - bt)|$$

#### Maximum Particle Speed

The maximum speed occurs when both sine terms reach their maximum value of $1$ simultaneously with matching signs:

$$v_{p,	ext{max}} = Ab + 0.8 Ab = 1.8 Ab$$

#### Minimum Particle Speed

The minimum speed occurs when both sine terms are zero simultaneously:

$$v_{p,	ext{min}} = 0$$

This shows that even with partial reflection, there are points in the medium where the particles can come to a complete standstill at certain instants.

### Part 5: The Elegant Decomposition – Standing vs. Travelling Waves

Any partial standing wave can be decomposed into a pure standing wave and a pure travelling wave.

Let's split the amplitude of the incident wave into two parts: $0.8 A$ (which will pair with the reflected wave to form a standing wave) and $0.2 A$ (which will remain as a travelling wave):

$$y = [0.8 A \cos(ax + bt) - 0.8 A \cos(ax - bt)] + 0.2 A \cos(ax + bt)$$

Now, let's apply the trigonometric identity:

$$\cos(C) - \cos(D) = -2 \sin\left(\frac{C+D}{2}ight) \sin\left(\frac{C-D}{2}ight)$$

For our terms, let $C = ax + bt$ and $D = ax - bt$:

$$\frac{C+D}{2} = ax \quad 	ext{and} \quad \frac{C-D}{2} = bt$$

Substituting these into the identity:

$$0.8 A [\cos(ax + bt) - \cos(ax - bt)] = -1.6 A \sin(ax) \sin(bt)$$

Therefore, the resultant wave can be written as:

$$y = -1.6 A \sin(ax) \sin(bt) + 0.2 A \cos(ax + bt)$$

This is an incredibly beautiful result!

- The term **$-1.6 A \sin(ax) \sin(bt)$** represents a **standing wave** with a spatial amplitude envelope of $1.6 A \sin(ax)$.
- The term **$0.2 A \cos(ax + bt)$** represents a **travelling wave** propagating in the **negative $x$-direction** (towards the left).

#### Finding the Antinodes

Antinodes are the positions where the amplitude of the standing wave component is maximum.

This occurs where:

$$|\sin(ax)| = 1$$

$$ax = (2n+1)\frac{\pi}{2} \implies x = (2n+1)\frac{\pi}{2a} \quad 	ext{for } n \in \mathbb{Z}$$

These are the positions of the antinodes!

### Conclusion: The Bigger Picture

This problem beautifully illustrates how partial reflection leads to a mixture of standing and travelling waves.

In real-world applications, such as architectural acoustics or transmission lines, minimizing this standing wave component (by matching impedances) is crucial to prevent energy loss and unwanted resonances.

By mastering these boundary conditions and trigonometric decompositions, you unlock a deeper understanding of wave physics that spans from acoustics to quantum mechanics!

Visualized Solution

Analyzing the Incident Wave

Wavelength and Frequency of Incident Wave

Boundary Conditions for Reflection

Equation of the Reflected Wave

Resultant Wave by Superposition

Particle Velocity Equation

Maximum and Minimum Particle Speeds

Standing and Travelling Wave Decomposition

Antinodes and Propagation Direction

Summary of Results

The Sigma Insight: Reflection and Transmission of Waves

Introduction

Part 1

Part 2

Part 3

Part 4

# Maximum Particle Speed

# Minimum Particle Speed

Part 5

# Finding the Antinodes

Conclusion

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