The displacement of particles in a string stretched in the x-direction is represented by y. Among th

Q: Why does a standing wave like cos(kx) sin(wt) count as wave motion if it doesn't travel anywhere?

Even though a standing wave doesn't propagate net energy, it is mathematically formed by the superposition of two identical progressive waves travelling in opposite directions. Since it satisfies the linear wave equation with a real wave speed $v = \omega/k$, it is classified as a valid wave motion.

Q: Why is option (b) incorrect if it satisfies the relation with a constant ratio?

While option (b) yields a constant ratio between its second derivatives, that ratio is negative. This implies $v^2 < 0$, which means the wave velocity $v$ would be an imaginary number. Since physical wave propagation speeds must be real, option (b) is physically impossible.

Q: Does any function of the form f(ax + bt) always represent a wave?

Yes, mathematically any twice-differentiable function of the form $f(ax \pm bt)$ satisfies the wave equation. However, physically, the function must also be defined and finite everywhere (i.e., it shouldn't blow up to infinity) to represent a real physical wave.

Q: Why does the squared argument in cos(k^2 x^2 - w^2 t^2) prevent it from being a wave?

When the arguments are squared, the second derivatives generate terms multiplied by $x^2$ and $t^2$. This makes the ratio of the spatial and temporal derivatives dependent on position and time, meaning there is no constant propagation speed $v$. Thus, it cannot represent a wave.

Q: How did we simplify cos^2(kx + wt) to show it is a wave?

We used the trigonometric identity $\cos^2 \theta = \frac{1 + \cos 2\theta}{2}$. This rewrites the function as $y = \frac{1}{2} + \frac{1}{2}\cos(2kx + 2\omega t)$, which is clearly of the form $f(ax + bt)$ with $a = 2k$ and $b = 2\omega$, confirming it is a progressive wave.

Question

The displacement of particles in a string stretched in the $x$-direction is represented by $y$. Among the following expressions for $y$, those describing wave motion is (are)

Options:

(A) $\cos kx \sin \omega t$

(B) $k^2 x^2 - \omega^2 t^2$

(C) $\cos^2(kx + \omega t)$

(D) $\cos(k^2 x^2 - \omega^2 t^2)$

Sigma Percentile · Accepted Answer

### Introduction to Wave Mathematics

Imagine standing by a quiet pond and dropping a single pebble into the water.

You instantly observe concentric ripples expanding outward, carrying energy across the surface.

In physics, we describe this beautiful phenomenon as a **wave**—a self-propagating disturbance that travels through a medium without causing any permanent displacement of the medium's particles themselves.

But how do we translate this physical beauty into rigorous mathematics?

Any physical wave, whether it is a sound wave in air, a light wave in vacuum, or a transverse wave on a stretched string, must satisfy a fundamental partial differential equation known as the **linear wave equation**:

$$\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$$

Here, $y(x,t)$ represents the displacement of the particles, $x$ is the position along the propagation axis, $t$ is time, and $v$ is the constant, real velocity of the wave.

In this article, we will put four different mathematical expressions to the test to see which ones represent true, physical wave motion.

---

### Analyzing Option (a): The Standing Wave

Let's begin our investigation with the first expression:

$$y = \cos kx \sin \omega t$$

This function represents a separation of spatial and temporal variables.

To see if it satisfies our master wave equation, let's calculate its second-order partial derivatives.

First, we differentiate with respect to position $x$:

$$\frac{\partial y}{\partial x} = -k \sin kx \sin \omega t$$

$$\frac{\partial^2 y}{\partial x^2} = -k^2 \cos kx \sin \omega t = -k^2 y$$

Next, we differentiate with respect to time $t$:

$$\frac{\partial y}{\partial t} = \omega \cos kx \cos \omega t$$

$$\frac{\partial^2 y}{\partial t^2} = -\omega^2 \cos kx \sin \omega t = -\omega^2 y$$

Now, let's substitute these back into our wave equation:

$$\frac{\partial^2 y}{\partial x^2} = \left(\frac{k^2}{\omega^2}ight) \frac{\partial^2 y}{\partial t^2}$$

Comparing this with the standard wave equation, we find that:

$$v^2 = \frac{\omega^2}{k^2} \implies v = \frac{\omega}{k}$$

Since the wave speed $v$ is a real, non-zero constant, this expression perfectly satisfies the wave equation!

Physically, this represents a **standing wave** (or stationary wave), which is formed by the superposition of two identical progressive waves travelling in opposite directions.

Therefore, **Option (a) is a correct choice**.

---

### Analyzing Option (b): The Quadratic Trap

Now, let's look at the second expression:

$$y = k^2 x^2 - \omega^2 t^2$$

At first glance, this polynomial form might seem like it could represent some sort of expanding disturbance.

Let's calculate its derivatives to find out:

$$\frac{\partial y}{\partial x} = 2k^2 x \implies \frac{\partial^2 y}{\partial x^2} = 2k^2$$

$$\frac{\partial y}{\partial t} = -2\omega^2 t \implies \frac{\partial^2 y}{\partial t^2} = -2\omega^2$$

If we try to relate these two constant derivatives using the wave equation, we get:

$$2k^2 = \frac{1}{v^2} (-2\omega^2) \implies v^2 = -\frac{\omega^2}{k^2}$$

Notice the negative sign!

This implies that the wave velocity $v$ must be an **imaginary number** ($v = i\frac{\omega}{k}$).

Since a physical wave cannot propagate with an imaginary speed, this expression does not represent a real wave motion.

Therefore, **Option (b) is incorrect**.

---

### Analyzing Option (c): The Progressive Wave

Let's move on to the third candidate:

$$y = \cos^2(kx + \omega t)$$

We can simplify this expression using the double-angle trigonometric identity:

$$\cos^2 	heta = \frac{1 + \cos 2	heta}{2}$$

Substituting this in, we get:

$$y = \frac{1}{2} [1 + \cos(2kx + 2\omega t)]$$

Notice that the entire spatial and temporal dependence of this function is contained within the single variable argument $u = kx + \omega t$.

Any function of the form $y = f(ax \pm bt)$ represents a progressive wave because its shape remains unchanged as it propagates.

Let's verify this mathematically by taking derivatives with respect to $u$:

$$\frac{\partial^2 y}{\partial x^2} = k^2 f''(u) \quad 	ext{and} \quad \frac{\partial^2 y}{\partial t^2} = \omega^2 f''(u)$$

Dividing these two equations yields:

$$\frac{\partial^2 y}{\partial x^2} = \left(\frac{k^2}{\omega^2}ight) \frac{\partial^2 y}{\partial t^2}$$

This perfectly matches the wave equation with a real, constant wave speed of:

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

This represents a **progressive wave** travelling in the negative $x$-direction.

Therefore, **Option (c) is a correct choice**.

---

### Analyzing Option (d): The Non-Linear Argument

Finally, let's examine the fourth expression:

$$y = \cos(k^2 x^2 - \omega^2 t^2)$$

Here, the arguments of space and time are squared inside the cosine function.

Let's find the first partial derivatives using the chain rule:

$$\frac{\partial y}{\partial x} = -2k^2 x \sin(k^2 x^2 - \omega^2 t^2)$$

$$\frac{\partial y}{\partial t} = 2\omega^2 t \sin(k^2 x^2 - \omega^2 t^2)$$

Now, let's find the second-order derivatives using the product rule:

$$\frac{\partial^2 y}{\partial x^2} = -2k^2 \sin(k^2 x^2 - \omega^2 t^2) - 4k^4 x^2 \cos(k^2 x^2 - \omega^2 t^2)$$

$$\frac{\partial^2 y}{\partial t^2} = 2\omega^2 \sin(k^2 x^2 - \omega^2 t^2) - 4\omega^4 t^2 \cos(k^2 x^2 - \omega^2 t^2)$$

Clearly, the ratio of these two second derivatives is not a constant; it depends heavily on the coordinates $x$ and $t$ due to the $x^2$ and $t^2$ terms.

Because there is no single, constant wave speed $v$ at which this disturbance propagates, this expression does not represent a valid wave motion.

Therefore, **Option (d) is incorrect**.

---

### Summary of Results

By testing each option against the classical wave equation, we have successfully separated the physical waves from the mathematical impostors:

*   **Option (a)** represents a valid **standing wave** with speed $v = \omega/k$.
*   **Option (c)** represents a valid **progressive wave** with speed $v = \omega/k$.

Thus, the correct options are **(a)** and **(c)**.

Animated Solution for Physics - Waves: The displacement of particles in a string stretched in the $x$ -direction is represented by $y$ . Among the following expressions for $y$ , those describing wave motion is (are)

Select Answer:

Visualized Solution

Understanding Wave Motion

The Linear Wave Equation

Testing Option (a): $y = cos k x sin ω t$

Verifying Option (a)

Testing Option (b): $y = k^{2} x^{2} - ω^{2} t^{2}$

Why Option (b) Fails

Testing Option (c): $y = cos^{2} (k x + ω t)$

Verifying Option (c)

Testing Option (d): $y = cos (k^{2} x^{2} - ω^{2} t^{2})$

Why Option (d) Fails

Final Conclusion

The Sigma Insight: Wave Equation and Wave Speed

Introduction to Wave Mathematics

Analyzing Option (a)

Analyzing Option (b)

Analyzing Option (c)

Analyzing Option (d)

Summary of Results

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