A wave equation which gives the displacement along the y-direction is given by : y = 10^-4 sin (60 t

Q: Why does a positive sign between the time and space terms mean the wave travels in the negative x-direction?

To maintain a constant phase (i.e., keeping the argument of the sine function constant), any increase in time $t$ must be balanced by a decrease in position $x$. Mathematically, if $\omega t + kx = C$, then differentiating with respect to time gives $\omega + k \frac{dx}{dt} = 0 \implies \frac{dx}{dt} = -\frac{\omega}{k}$, which is negative.

Q: Is wave velocity the same as particle velocity?

No, they are entirely different. Wave velocity ($v = \omega/k$) is the speed at which the wave profile (or energy) propagates through the medium. Particle velocity ($v_p = \partial y/\partial t$) is the speed at which individual particles of the medium oscillate about their mean positions. In this wave, the wave speed is $30\text{ m/s}$, while the maximum particle speed is only $A\omega = 0.006\text{ m/s}$.

Q: What is the physical meaning of the wave number k?

The wave number $k = 2\pi/\lambda$ represents the spatial frequency of the wave. It tells us how many radians of phase change occur per unit distance along the direction of propagation.

Q: Does the phase constant affect the speed or wavelength of the wave?

No. The phase constant $\phi$ only shifts the wave in space and time (determining where the crests and troughs are at $t=0, x=0$). It has absolutely no effect on intrinsic properties like wave speed, wavelength, frequency, or amplitude.

Q: How do we find the maximum acceleration of a particle in this wave?

Since the particles execute Simple Harmonic Motion (SHM) perpendicular to the direction of wave propagation, their maximum acceleration is given by $a_{p,\text{max}} = \omega^2 A$. For this wave, $a_{p,\text{max}} = (60)^2 \times 10^{-4} = 0.36\text{ m/s}^2$.

Question

A wave equation which gives the displacement along the $y$-direction is given by : $y = 10^{-4} \sin (60 t + 2x)$ where, $x$ and $y$ are in metre and $t$ is time in second. This represents a wave

Options:

(A) travelling with a velocity of $30	ext{ m/s}$ in the negative $x$-direction

(B) of wavelength $(\pi)	ext{ m}$

(C) of frequency $\left(\frac{30}{\pi}ight)	ext{ Hz}$

(D) of amplitude $10^{-4}	ext{ m}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a string stretched out infinitely along the $x$-axis.

When you wiggle one end of this string, you create a disturbance that travels down its length.

This travelling disturbance is what we mathematically call a **progressive wave**.

In this problem, we are handed a specific mathematical description of such a wave:

$$y = 10^{-4} \sin (60 t + 2x)$$

Here, $y$ represents the vertical displacement of any particle on the string at position $x$ and time $t$.

Our mission is to dissect this equation and extract all its physical secrets: its amplitude, its wavelength, its frequency, its speed, and the direction in which it is marching.

---

### The Master Equation

To unlock the physical parameters hidden inside our wave equation, we must compare it with the standard mathematical template of a progressive wave:

$$y = A \sin(\omega t + kx + \phi)$$

Let's break down what each of these symbols represents:

*   **$A$ (Amplitude):** The maximum height of the wave crest or depth of the wave trough. It tells us how much energy the wave carries.
*   **$\omega$ (Angular Frequency):** This represents how fast the wave oscillates in time, measured in radians per second.
*   **$k$ (Wave Number):** This is the spatial counterpart of angular frequency. It tells us how many wave cycles fit into a unit of space.
*   **$\phi$ (Phase Constant):** This determines the initial state of the wave at $t=0$ and $x=0$.

By directly comparing our given equation $y = 10^{-4} \sin(60t + 2x)$ with the standard form, we can immediately read off the following values:

1.  **Amplitude ($A$):** $10^{-4} 	ext{ m}$
2.  **Angular Frequency ($\omega$):** $60 	ext{ rad/s}$
3.  **Wave Number ($k$):** $2 	ext{ m}^{-1}$

This simple comparison immediately validates **Option (d)**, which states that the amplitude is $10^{-4} 	ext{ m}$.

---

### Direction of Propagation

Before we dive into calculations, let's address a crucial conceptual question: **Which way is the wave travelling?**

Look closely at the terms inside the sine function: $60t$ and $2x$.

Both terms have the **same sign** (both are positive).

In wave mechanics, if the coefficients of $t$ and $x$ have the same sign, the wave propagates in the **negative $x$-direction** (to the left).

Why is this the case?

To keep the phase of the wave constant (i.e., to stay on the same point of the wave crest as time $t$ increases), the position $x$ must decrease.

Conversely, if the signs were opposite (e.g., $60t - 2x$), the wave would travel in the **positive $x$-direction** (to the right).

Therefore, our wave is travelling in the **negative $x$-direction**.

---

### Calculating Wavelength and Frequency

Now, let's compute the spatial and temporal characteristics of the wave.

#### 1. Wavelength ($\lambda$)

The wave number $k$ is related to the wavelength $\lambda$ by the fundamental formula:

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

Substituting our extracted value of $k = 2$:

$$2 = \frac{2\pi}{\lambda} \implies \lambda = \pi 	ext{ m}$$

This confirms that **Option (b)** is correct.

#### 2. Frequency ($f$)

The angular frequency $\omega$ is related to the linear frequency $f$ (the number of full cycles per second) by:

$$\omega = 2\pi f$$

Substituting our extracted value of $\omega = 60$:

$$60 = 2\pi f \implies f = \frac{30}{\pi} 	ext{ Hz}$$

This confirms that **Option (c)** is correct.

---

### Calculating Wave Velocity

Finally, let's find out how fast this wave is travelling through space.

The velocity of a progressive wave ($v$) is given by the ratio of its angular frequency to its wave number:

$$v = \frac{\omega}{k}$$

Substituting our values:

$$v = \frac{60}{2} = 30 	ext{ m/s}$$

Alternatively, we can use the classic wave speed formula relating frequency and wavelength:

$$v = f \lambda = \left(\frac{30}{\pi}ight) 	imes \pi = 30 	ext{ m/s}$$

Since the wave is travelling in the negative $x$-direction, its velocity vector is directed along the negative $x$-axis with a magnitude of $30 	ext{ m/s}$.

This perfectly matches **Option (a)**.

---

### Conclusion

By systematically breaking down the wave equation, we have verified that:

*   The wave travels with a velocity of $30 	ext{ m/s}$ in the negative $x$-direction (**Option a is correct**).
*   The wavelength of the wave is $\pi 	ext{ m}$ (**Option b is correct**).
*   The frequency of the wave is $\frac{30}{\pi} 	ext{ Hz}$ (**Option c is correct**).
*   The amplitude of the wave is $10^{-4} 	ext{ m}$ (**Option d is correct**).

This is a beautiful example of a multi-correct JEE question where **all four options (a, b, c, d) are correct**!

Animated Solution for Physics - Waves: A wave equation which gives the displacement along the $y$ -direction is given by : $y = 1 0^{- 4} sin (60 t + 2 x)$ where, $x$ and $y$ are in metre and $t$ is time in second. This represents a wave

Select Answer:

Visualized Solution

The Wave Equation

The Standard Wave Equation

Extracting Wave Parameters

Direction of Propagation

Calculating Wavelength ( $λ$ )

Calculating Frequency ( $f$ )

Calculating Wave Velocity ( $v$ )

Final Synthesis

The Way Forward

The Sigma Insight: Wave Equation and Wave Speed

Analyzing the Setup

The Master Equation

Direction of Propagation

Calculating Wavelength and Frequency

# 1

# 2

Calculating Wave Velocity

Conclusion

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Visualized Solution

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