The displacement y of a particle in a medium can be expressed as y = 10^-6 sin(100t + 20x + pi/4) m

Q: Why is the wave speed calculated as $\omega/k$?

Wave speed is fundamentally $v = f\lambda$. Since angular frequency $\omega = 2\pi f$ and wave number $k = \frac{2\pi}{\lambda}$, dividing them gives $\frac{\omega}{k} = \frac{2\pi f}{\frac{2\pi}{\lambda}} = f\lambda = v$.

Q: Does the phase angle $\frac{\pi}{4}$ affect the wave speed?

No, the initial phase angle only shifts the wave in space or time at $t=0$. It determines the starting position of the oscillation but does not affect how fast the wave propagates through the medium.

Q: What does the positive sign between $100t$ and $20x$ indicate?

A positive sign in the argument $(\omega t + kx)$ indicates that the wave is travelling in the negative x-direction. If it were $(\omega t - kx)$, it would be travelling in the positive x-direction.

Q: What is the amplitude of this wave?

The amplitude is the coefficient of the sine function, which represents the maximum displacement from the mean position. Here, it is $10^{-6}$ meters.

Q: Is this a transverse or longitudinal wave?

The equation describes the displacement $y$ as a function of position $x$. This is typical for a transverse wave where the particle displacement ($y$-axis) is perpendicular to the direction of wave propagation ($x$-axis).

Question

The displacement $y$ of a particle in a medium can be expressed as
$y = 10^{-6} \sin \left( 100t + 20x + \frac{\pi}{4} ight) 	ext{m}$
where, $t$ is in second and $x$ in metre. The speed of the wave is

Options:

(A) $2000 	ext{ ms}^{-1}$

(B) $5 	ext{ ms}^{-1}$

(C) $20 	ext{ ms}^{-1}$

(D) $5\pi 	ext{ ms}^{-1}$

Sigma Percentile · Accepted Answer

### Decoding the Speed of a Travelling Wave

Imagine you are standing by a calm pond and you toss a pebble into it. You see ripples moving outward. That moving ripple is a wave, and just like a car on a highway, it has a specific speed. In physics, we describe this beautiful motion using a mathematical equation. Let's break down how to extract the speed of a wave just by looking at its equation.

### The Anatomy of a Wave Equation

The problem gives us the displacement $y$ of a particle in a medium as:

$$y = 10^{-6} \sin \left( 100t + 20x + \frac{\pi}{4} ight)$$

At first glance, this might look like a jumble of numbers and variables. But every single piece of this equation tells a story about the wave's physical reality. To decode it, we bring in our master key—the standard equation of a progressive travelling wave:

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

Let's map the parts:
- **$a$** is the amplitude, the maximum height of the wave. Here, it's a tiny $10^{-6} 	ext{ m}$.
- **$\omega$** is the angular frequency, which tells us how fast the particles are oscillating up and down. By comparing the equations, we see $\omega = 100 	ext{ rad s}^{-1}$.
- **$k$** is the angular wave number, which tells us how densely packed the waves are in space. Comparing again, we find $k = 20 	ext{ m}^{-1}$.
- **$\phi$** is the initial phase, which just tells us where the wave cycle started at $t=0$. Here, it's $\frac{\pi}{4}$.

### The Master Equation for Wave Speed

Now, how do we find the speed at which the wave pattern itself is travelling forward? The wave speed $v$ is beautifully simple. It is the ratio of how fast the wave oscillates in time ($\omega$) to how fast it oscillates in space ($k$).

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

Why does this work? Remember that $\omega = 2\pi f$ (where $f$ is frequency) and $k = \frac{2\pi}{\lambda}$ (where $\lambda$ is wavelength). If you divide them, the $2\pi$ cancels out, leaving $f 	imes \lambda$, which is the classic formula for wave speed!

### Final Calculation

Let's plug in our extracted values:

$$v = \frac{100}{20}$$

$$v = 5 	ext{ ms}^{-1}$$

And there we have it! The wave is travelling through the medium at a crisp speed of $5 	ext{ ms}^{-1}$. By simply comparing coefficients, we unlocked the physical behavior of the wave.

Animated Solution for Physics - Waves: The displacement $y$ of a particle in a medium can be expressed as $y = 1 0^{- 6} sin (100 t + 20 x + \frac{π}{4}) m$ where, $t$ is in second and $x$ in metre. The speed of the wave is

Select Answer:

Visualized Solution

y = 10^{-6} \sin \left( 100t + 20x + \frac{\pi}{4} \right)

y = a \sin (\omega t + kx + \phi)

\omega = 100 \text{ rad s}^{-1}, k = 20 \text{ m}^{-1}

v = \frac{\omega}{k}

v = 5 \text{ ms}^{-1}

The Sigma Insight: Wave Equation and Wave Speed

Decoding the Speed of a Travelling Wave

The Anatomy of a Wave Equation

The Master Equation for Wave Speed

Final Calculation

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