Y(x, t) = 0.8 / [(4x + 5t)^2 + 5] represents a moving pulse where x and y are in metre and t is in s

Q: Why does a positive sign in $(4x + 5t)$ mean the wave is moving in the negative x-direction?

For a wave profile to maintain its shape as time $t$ increases, any constant phase point must satisfy $4x + 5t = \text{constant}$. Differentiating this with respect to time gives $4\frac{dx}{dt} + 5 = 0$, which yields a negative velocity $\frac{dx}{dt} = -1.25\text{ m/s}$. Hence, the wave moves in the negative $x$-direction.

Q: Can the maximum displacement ever be greater than 0.16 m at other times?

No. The maximum displacement is determined solely by minimizing the denominator. Since the term $(4x + 5t)^2$ is squared, its minimum value is always $0$ regardless of the values of $x$ and $t$. Thus, the maximum displacement remains exactly $0.16\text{ m}$ at all times; the peak just shifts in space.

Q: How do we define a symmetric pulse mathematically?

A pulse is symmetric if its spatial profile at any frozen instant (usually $t=0$) is an even function, meaning $Y(x) = Y(-x)$ when the peak is centered at $x=0$. Since our function at $t=0$ depends only on $x^2$, it is perfectly symmetric.

Q: What is the difference between wave velocity and particle velocity here?

Wave velocity ($1.25\text{ m/s}$) is the speed at which the pulse profile propagates through the medium. Particle velocity is the speed at which individual elements of the string move vertically, given by the partial derivative $\frac{\partial Y}{\partial t}$.

Q: Is this wave a transverse wave or a longitudinal wave?

Since $Y$ represents displacement and is a function of $x$ and $t$, if $Y$ is perpendicular to the direction of propagation ($x$-axis), it is a transverse wave. The problem statement implies standard transverse string pulse representation.

Question

A moving pulse is represented by the wave function $Y(x, t) = \frac{0.8}{(4x + 5t)^2 + 5}$, where $x$ and $y$ are in metre and $t$ is in second. Then,

Options:

(A) pulse is moving in positive $x$-direction

(B) in $2	ext{ s}$ it will travel a distance of $2.5	ext{ m}$

(C) its maximum displacement is $0.16	ext{ m}$

(D) it is a symmetric pulse

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a localized disturbance—a **wave pulse**—travelling along a stretched string.

Mathematically, we represent this moving shape using a wave function $Y(x, t)$ that depends on both space $x$ and time $t$.

In this problem, we are handed a specific mathematical model for our pulse:

$$Y(x, t) = \frac{0.8}{(4x + 5t)^2 + 5}$$

Our mission is to dissect this equation and uncover the physical properties of the pulse: its direction of motion, its speed, its maximum height, and its symmetry. Let's dive in!

### The Direction of Propagation

To find out which way the wave is moving, we look at the argument inside the function.

Any progressive wave travelling along a straight line can be written in the general form:

$$y = f(ax \pm bt)$$

Here, the sign between the spatial term $ax$ and the temporal term $bt$ is the ultimate compass:
- A **negative sign** ($ax - bt$) means the wave is moving in the **positive $x$-direction**.
- A **positive sign** ($ax + bt$) means the wave is moving in the **negative $x$-direction**.

Looking at our wave function, the argument is $(4x + 5t)$.

Since the sign between $4x$ and $5t$ is positive, the pulse is propagating in the **negative $x$-direction**.

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

### Calculating Wave Speed and Distance

How fast is this pulse travelling?

The speed of propagation $v$ of any wave of the form $f(ax + bt)$ is given by the ratio of the coefficients:

$$v = \frac{	ext{Coefficient of } t}{	ext{Coefficient of } x} = \frac{b}{a}$$

Substituting our values:

$$v = \frac{5}{4} = 1.25 	ext{ m/s}$$

Now, let's calculate the distance $d$ travelled by the pulse in a time interval of $t = 2 	ext{ s}$:

$$d = v 	imes t = 1.25 	imes 2 = 2.5 	ext{ m}$$

This matches option (b) perfectly!

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

### Finding the Peak of the Pulse

What is the maximum displacement of the string?

To find the maximum value of $Y(x, t)$, we need to look at the fraction:

$$Y(x, t) = \frac{0.8}{(4x + 5t)^2 + 5}$$

To make $Y$ as large as possible, we must make the denominator as small as possible.

The denominator consists of a squared term $(4x + 5t)^2$ and a constant $5$.

Since the square of any real number is always non-negative, the absolute minimum value of $(4x + 5t)^2$ is $0$, which occurs when:

$$4x + 5t = 0$$

At this peak condition, the denominator reaches its minimum value of $5$.

Thus, the maximum displacement $Y_{\max}$ is:

$$Y_{\max} = \frac{0.8}{5} = 0.16 	ext{ m}$$

This confirms option (c) is correct!

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

### Verifying Symmetry

Finally, let's check if the pulse is symmetric.

To analyze the shape of the pulse, we can freeze time at $t = 0$:

$$Y(x, 0) = \frac{0.8}{(4x)^2 + 5} = \frac{0.8}{16x^2 + 5}$$

A function is symmetric about $x = 0$ if replacing $x$ with $-x$ yields the exact same function:

$$Y(-x, 0) = \frac{0.8}{16(-x)^2 + 5} = \frac{0.8}{16x^2 + 5} = Y(x, 0)$$

Because the spatial variable $x$ is squared, the negative sign vanishes completely.

This proves that the pulse is perfectly symmetric about its peak.

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

### Conclusion

By systematically breaking down the wave function, we have determined that:
1. The wave travels in the **negative $x$-direction**.
2. The wave speed is **$1.25	ext{ m/s}$**, covering **$2.5	ext{ m}$** in **$2	ext{ s}$**.
3. The maximum displacement is **$0.16	ext{ m}$**.
4. The pulse shape is **perfectly symmetric**.

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

Animated Solution for Physics - Waves: A moving pulse is represented by the wave function $Y (x, t) = \frac{0.8}{( 4 x + 5 t ) ^{2} + 5}$ , where $x$ and $y$ are in metre and $t$ is in second. Then,

Select Answer:

Visualized Solution

Understanding the Wave Pulse Representation

Determining the Direction of Propagation

Calculating the Wave Velocity

Computing the Speed and Distance

Finding the Maximum Displacement

Calculating $Y_{m a x}$

Checking for Pulse Symmetry

Verifying $Y (x, 0) = Y (- x, 0)$

Summary of Correct Options

The Sigma Insight: Wave Equation and Wave Speed

Analyzing the Setup

The Direction of Propagation

Calculating Wave Speed and Distance

Finding the Peak of the Pulse

Verifying Symmetry

Conclusion

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