Which of the following equations represents a travelling wave?
(a) y = A sin(15x - 2t)
(b) y = A e^(

Q: Why must a travelling wave be of the form $f(ax \pm bt)$?

For a wave to maintain its shape while travelling at a constant speed $v$, the displacement at position $x$ and time $t$ must be the same as the displacement at position $x - vt$ at time $t=0$. This naturally leads to the argument being proportional to $(x \pm vt)$, or $(kx \pm \omega t)$.

Q: What does option (d) represent physically?

Option (d), $y = A \sin x \cos \omega t$, represents a standing wave. In a standing wave, the profile doesn't move through space; instead, the amplitude of oscillation varies with position $x$, creating fixed nodes (zero amplitude) and antinodes (maximum amplitude).

Q: Can a travelling wave have an exponential form?

Yes! A pulse like $y = A e^{-(x-vt)^2}$ is a perfectly valid travelling wave. It represents a bell-shaped pulse moving with speed $v$. The key is the $(x-vt)$ argument, which option (b) lacks.

Q: How do we find the speed of the wave from the equation $y = A \sin(15x - 2t)$?

The speed $v$ is given by the ratio of the coefficient of $t$ to the coefficient of $x$. Here, $\omega = 2$ and $k = 15$, so the speed is $v = \frac{\omega}{k} = \frac{2}{15}$ m/s.

Q: What does the minus sign in $(15x - 2t)$ indicate?

The minus sign indicates that the wave is travelling in the positive $x$-direction. If it were $(15x + 2t)$, the wave would be travelling in the negative $x$-direction.

Question

Which of the following equations represents a travelling wave?

Options:

(A) $y = A \sin(15x - 2t)$

(B) $y = A e^{-x^2} (vt + 	heta)$

(C) $y = A e^x \cos(\omega t - 	heta)$

(D) $y = A \sin x \cos \omega t$

Sigma Percentile · Accepted Answer

### The Mathematical Anatomy of a Travelling Wave

Imagine standing by a calm pond and tossing a pebble into the center. You watch as a circular ripple expands outward, maintaining its shape as it glides across the surface of the water. This beautiful, translating disturbance is what physicists call a **travelling wave**. But how do we capture this dynamic, moving reality into a static mathematical equation?

The secret lies in how space ($x$) and time ($t$) are locked together. For a wave to travel without distorting its shape, the displacement $y(x,t)$ must depend on a very specific linear combination of position and time: $(ax \pm bt)$ or $(kx \pm \omega t)$. If you shift your position by a certain amount, and wait a proportional amount of time, the wave looks exactly the same. This locked argument is the absolute hallmark of a travelling wave.

### Analyzing the Contenders

Let's put our options to the test using this strict mathematical filter.

Look at **Option (a)**: $$y = A \sin(15x - 2t)$$
Notice the argument inside the sine function: $(15x - 2t)$. This is a perfect match for our required form $(ax - bt)$, where $a = 15$ and $b = 2$. Because the signs of the $x$ and $t$ terms are opposite, this equation describes a wave marching confidently in the positive $x$-direction. We have found our travelling wave!

But what about the others? Why do they fail?

In **Option (b)**, $y = A e^{-x^2} (vt + 	heta)$, the spatial part (the exponential) and the temporal part are multiplied together, but they are not locked inside a single argument. The shape of this disturbance will change over time; it won't just cleanly translate.

Similarly, in **Option (c)**, $y = A e^x \cos(\omega t - 	heta)$, the position $x$ is trapped in an exponential, while time $t$ is oscillating inside a cosine. They are completely divorced from one another. This is not a travelling wave.

### The Standing Wave Trap

Finally, we arrive at **Option (d)**: $$y = A \sin x \cos \omega t$$
This is a classic trap. Here, the spatial harmonic function ($\sin x$) and the temporal harmonic function ($\cos \omega t$) are completely separated. This equation represents a **standing wave** (or stationary wave).

In a standing wave, the energy doesn't travel from left to right. Instead, the wave simply oscillates up and down in place. There are fixed points called nodes that never move, and antinodes that bounce between maximum positive and negative amplitudes. It's a beautiful phenomenon, but it is strictly stationary, not travelling.

By understanding the deep connection between the physical motion of a wave and its mathematical argument $(kx \pm \omega t)$, you can instantly identify the true nature of any wave equation you encounter.

Animated Solution for Physics - Waves: Which of the following equations represents a travelling wave?

Select Answer:

Visualized Solution

Concept of a Travelling Wave

Analyzing Option (a)

Analyzing Option (b) \& (c)

Analyzing Option (d)

Conclusion

The Way Forward

The Sigma Insight: Wave Equation and Wave Speed

The Mathematical Anatomy of a Travelling Wave

Analyzing the Contenders

The Standing Wave Trap

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