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A Wave Pulse | What Is A Wave Pulse

A Wave Pulse | What Is A Wave Pulse :- A wave pulse is a single disturbance that travels through a medium due to the vibration of its particles, and ideally it retains its shape as it propagates. If you hold a rope and give it one quick jerk, the bump that travels along the rope is a wave pulse. If you keep moving your hand repeatedly, you produce a continuous wave instead.

Unlike continuous waves that repeat again and again, a wave pulse represents a single, short-lived disturbance that travels through a medium, carrying energy without transporting matter.

Mathematical Representation of a Wave Pulse

A travelling wave pulse can be represented as :

$\displaystyle y=\frac{a}{b+{{\left( x\mp \upsilon t \right)}^{2}}}$ …..(1)

This equation describes a wave pulse moving without change in shape.

In expression (1), is displacement, position, time, and $\displaystyle \upsilon =\frac{coefficient\text{ }of\text{ }t}{coefficient\text{ }of\text{ }x}$ the speed of the pulse. The constant affects the height and the width, with the amplitude of the pulse equal to (maximum at ). In the standard form of a travelling pulse, the term appears as with the coefficient of equal to 1.

Why this expression represents a wave pulse

A wave pulse must be :

(1). Localized (exists only in a small region)
(2). Single disturbance
(3). Moves without changing shape

This equation satisfies all three.

(1). Why it is a localized pulse

Look at denominator :

$\displaystyle b+{{\left( x\mp \upsilon t \right)}^{2}}$

When is far from :

So disturbance is only significant near

That’s exactly what a pulse does — one bump moving.

(2). Single hump

The function produces one smooth peak, not repeating waves.

(3). Moves without changing shape

For a travelling pulse

$\displaystyle y=f\left( x-\upsilon t \right)$

The peak occurs when , giving

$\displaystyle {{y}_{max}}=f\left( 0 \right)$

which is constant, so the peak height does not change.

Now take any point at a distance from the peak :

$\displaystyle y=f\left( x-\upsilon t \right)=f\left( \upsilon t+d-\upsilon t \right)=f\left( d \right)$

This value depends only on d, not on time. So every point keeps the same displacement relative to the peak.

Therefore, the pulse shifts in position but its shape remains unchanged.

Example 1.

The wave function of a pulse is given by $\displaystyle y=\frac{5}{1+{{\left( 4x+6t \right)}^{2}}}$ , x and y are in meter and t is in seconds. Determine the pulse velocity and indicate the direction of propagation of pulse.

Solution :

Here

$\displaystyle y=\frac{5}{1+{{\left( 4x+6t \right)}^{2}}}$

Standard form of a travelling pulse,

$\displaystyle \Rightarrow y=\frac{5}{1+16{{\left( x+\frac{6}{4}t \right)}^{2}}}=\frac{5}{1+16{{\left( x+1.5t \right)}^{2}}}$

Hence velocity of pulse,

$\displaystyle \upsilon =1.5m/s$

$\displaystyle \upsilon =\frac{coefficient\text{ }of\text{ }t}{coefficient\text{ }of\text{ }x}=\frac{6}{4}=1.5m/s$

Since there occurs (+) sign between x and t, ➡ Pulse moves to the left.

Example 2.

$\displaystyle y(x,t)=\frac{1.2}{\left[ {{\left( 3x-6t \right)}^{2}}+4 \right]}$ , represents a moving pulse where are in meters and in seconds.

Select the correct alternative(s) :

(A) Pulse is moving in negative x-direction
(B) In 3 s it will travel a distance of 6 m
(C) Its maximum displacement is 0.3 m
(D) It is a symmetric pulse

Solution :

Put into standard form :

$\displaystyle y(x,t)=\frac{1.2}{\left[ 9{{\left( x-\frac{6}{3}t \right)}^{2}}+4 \right]}=\frac{1.2}{\left[ 9{{\left( x-2t \right)}^{2}}+4 \right]}$

Since it is ( → motion in positive x-direction.

So (A) ❌

Distance travelled in 3 s,

Distance = vt = 2 × 3 = 6m

(B) ✅

Maximum displacement (Amplitude),

$\displaystyle {{y}_{max}}=\frac{a}{b}=\frac{1.2}{4}=0.3m$

Symmetry,

A pulse is symmetric (about its maximum displacement if, at , its displacement satisfies y(x) = y(-x).

From the given equation :

$\displaystyle \left. \begin{array}{l}y(x)=\frac{1.2}{\left[ 9{{x}^{2}}+4 \right]}\\y(-x)=\frac{1.2}{\left[ 9{{x}^{2}}+4 \right]}\end{array} \right\}\text{ }at\text{ }t=0$

$\displaystyle \Rightarrow y(x)=y(-x)$

Therefore, pulse is symmetric.

(D) ✅

Hence correct options are (B), (C) and (D).

A Wave Pulse | What Is A Wave Pulse

Mathematical Representation of a Wave Pulse

Why this expression represents a wave pulse

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