A light wire is tightly stretched with tension F. Transverse traveling waves of amplitude A and wavelength A₁carry average power Pavg = 0.400 W. If the wavelength of the waves is doubled, so A₂= 2A₁, while the tension F and amplitude A are not altered, what then is the average power Pav,2 carried by the waves?

Short Answer

Expert verified

The average power carried by the waves is Pavg,2=0.100W.

Step by step solution

01

Determination of the formula of Mechanical Waves

The average power carried by a sinusoidal wave on a string is given by:

P=12μA2ω2v --(1)

The speed of a periodic wave u with wavelength and frequency f is given by:

v=f×λ --(2)

And the relation between the angular speed of a wave and its frequency is:

w=2πf--(3)

02

Step 2:Application of the formula of Mechanical Waves

First, rearrange equation (2), so we get a relation between the frequency and the wavelength:

f=vλ

Substitute for f into relation (3), we get:

ω=2πvλ

Now,substitute for w into equation (1), so we get:

Pavg=12μT2πvλ2A2

The average power carried by the wave is inversely proportional to the square of its wavelength:

role="math" localid="1664345158373" Pavgα1λ2Pavg,2Pavg,1=λ12λ22



So, when the wavelength is doubled (λ₂= 2λ₁), the average power becomes:

Pavg,2Pavg,1=λ122λ12=14Pavg,2=Pavg,14

Now, put in the value for Pavg,1 = 0.400W, so we get:

Pavg,2=0.400W4=0.100WPavg,2=0.100W

Therefore, the average power carried by the waves is Pavg,2=0.100W.

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