Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 11, Problem 90P
To determine

The amplitude satisfy the equation A=2Acos(ωt).

Expert Solution & Answer
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Answer to Problem 90P

It is proved that amplitude satisfy the equation A=2Acos(ωt), where, A is the amplitude of the wave plotted.

Explanation of Solution

Consider the expression for y as given below

y=A[sin(kxωt)+sin(kx+ωt)]=Asin(kxωt)+Asin(kx+ωt) (I)

Use the trigonometric relation given below to simplify equation (I)

sinα+sinβ=2sin[(α+β)2]cos[(αβ)2] (II)

Use equation (II) in (I)

y=Asin(kxωt)+Asin(kx+ωt)=2Asin(kxωt+kx+ωt2)cos(kxωt(kx+ωt)2)=2Asin(kx)cos(ωt)=[2Acos(ωt)]sin(kx) (III)

Substitute A for 2Acos(ωt) in equation (III)

y=Asin(kx) (IV)

Conclusion:

Let us check whether 2Acos(ωt) holds each values of amplitudes of the graph

Substitute 5.0cm for A, π5.0rad/cm for k, π6.0rad/s in equation (II)

y(x,t)=(5.0cm){sin((π5.0rad/cm)x(π6.0rad/s)t)+sin((π5.0rad/cm)x+(π6.0rad/s)t)}=2(5.0cm)sin((π5.0rad/cm)x)cos((π6.0rad/s)t)=(10.0cm)cos[(π6.0rad/s)t]sin[(π5.0rad/cm)x]

Thus, the maximum value of y is 10.0cm for x=2.5cm, and t=0.

For t=1.0s, the value of y(x,t) is

y(2.5cm,1.0s)=(10.0cm)cos[(π6.0rad/s)(1.0s)]sin[(π5.0rad/cm)(2.5cm)]=(10.0cm)cos[(π6.0rad/s)(1.0s)]×1=8.7cm

For t=2.0s

y(2.5cm,1.0s)=(10.0cm)cos[(π6.0rad/s)(2.0s)]sin[(π5.0rad/cm)(2.5cm)]=(10.0cm)cos[(π6.0rad/s)(2.0s)]×1=5.0cm

These values are correctly matching with those in graph.

In case of original function, for t=1.0s

y=(5.0cm)[sin((π5.0rad/cm)(2.5cm)(π6.0rad/s)(1.0s))+sin((π5.0rad/cm)(2.5cm)+(π6.0rad/s)(1.0s))]=8.7cm

For t=2.0s

y=(5.0cm)[sin((π5.0rad/cm)(2.5cm)(π6.0rad/s)(2.0s))+sin((π5.0rad/cm)(2.5cm)+(π6.0rad/s)(2.0s))]=5.0cm

Therefore amplitudes in graph satisfies equation A=2Acos(ωt), where A is the amplitude of the wave plotted

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