From Eq. [x = 0, , , , …..], the instantaneous rate at w…

Physics Wave and Sound Mix MCQ (Single Correct)

From Eq. [x = 0, , , , …..], the instantaneous rate at which a wave transmits energy along a string (instantaneous power) is

P(x, t) = –F

where F is the tension.

Evaluate P(x, t) for a standing wave of the form given by Eq.[y(x, t)= (Aω sin kx) sin ωt].

Show that for all values of x, the average power P av carried by the standing wave is zero. (Equation [p av =

ω 2 A 2 ] does not apply here. Can you see why?)

For a standing wave given by Eq. [y(x, t)= (Aω sin kx) sin ωt], graph P(x, t) and the displacement y(x, t) as functions of x for t = 0, t = 𝜋 /4ω, t = 𝜋 /2ω, and t = 3 𝜋 /4ω. (Positive P(x, t) means energy is flowing in the +x-direction; negative P(x, t) means the flow is in the –x-direction.)

The kinetic energy per unit length of string is greatest where the string has the greatest transverse speed, and the potential energy per unit length of string is greatest where the string has the steepest slope (because there the string is stretched the most). Using these ideas, discuss the flow of energy along the string.

Text Solution

Verified by Experts

The correct answer is:

CHECK THE SOLUTION.

= kA SW coskx sin ωt, = – ωA SW ωsinkx cosωt, and so the instantaneous power is

P = FA 2 SW ωk(sin kx cos kx)(sin ωt cos ωt)

= FA 2 SW ωk sin(2kx)sin(2ωt).

The average value of P is proportional to the average value of sin (2ωt), and the average of the since function is zero; P av = 0.

The waveform is the solid line, and the power is the dashes line. At time t = 𝜋 /2ω, y = 0 and P = 0 and the graphs coincide.

When the standing wave is at its maximum displacement at all points, all of the energy is potential, and is concentrated at the places where the slope is steepest (the nodes). When the standing wave has zero displacement, all of the energy is kinetic, concentrated where the particles are moving the fastest (the antinodes). Thus, the energy must be transferred from the nodes to the antinodes, and back again, twice in each cycle. Note that |P| is greatest midway between adjacent nodes and antinodes, and that P vanishes at the nodes and antinodes.