Two transverse sinusoidal waves combining in a medium are described by the wave functions y_1 =...

Question:

Two transverse sinusoidal waves combining in a medium are described by the wave functions

{eq}y_1 = {/eq} 1.00 sin π (x+0.700t)

{eq}y_2 = {/eq} 1.00sin π (x+0.700t)

Where x, {eq}y_1 {/eq}, and {eq}y_2 {/eq} are in centimeters and t is in seconds. Determine the maximum transverse position of an element of the medium at the following positions.

a) x=0.160 cm

{eq}|y_{max}|= {/eq}

b) x=0.360 cm

{eq}|y_{max}|= {/eq}

c) x=1.30 cm

{eq}|y_{max}|= {/eq}

d) Find the three smallest values of x corresponding to antinodes.

(Enter your answer from smallest to largest.)

Transverse Wave:

in this type of wave vibration of the medium particles take place perpendicular to the direction of motion. If the motion of the particles takes place sinusoidally it is call simple harmonic transverse wave. In a progressive wave, wave front moves in the medium .

Answer and Explanation:

given:

{eq}y_1(x,t)=1.00sin \pi (x+0.700t) \\ y_2(x,t)=1.00sin \pi (x-0.700t) \\ {/eq}

resultant displacement

{eq}y(x,t)=y_1(x,t)+y_2(x,t) \\ =1.00sin \pi (x+0.700t) +1.00sin \pi (x-0.700t) \\ =2.00 sin \pi x cos \pi 0.7t \\ for\ y_{max} , cos\pi 0.7t=1 {/eq}

a)

{eq}y_{max} (at\ x=0.16\ cm) =2.00 sin \pi *0.16 = 0.96 cm {/eq}

b)

{eq}y_{max} (at\ x=0.36\ cm) =2.00 sin \pi *0.36 = 1.8\ cm {/eq}

c)

{eq}y_{max} (at\ x=0.36\ cm) =2.00 sin \pi *0.36 cos \pi 0.7t = 1.62\ cm {/eq}

d)

condition for antinodes:

{eq}sin\pi x=\pm1 \Rightarrow \pi x= n \dfrac{\pi }{2} \\ \Rightarrow x=\dfrac{n}{2} {/eq}

where n=1,3,5,7..........

{eq}n=1, x_1= \dfrac{1}{2} =0.5\ cm \\ n=3, x_2= \dfrac{3}{2} =1.5\ cm \\ n=5, x_2= \dfrac{7}{2} =3.5\ cm {/eq}


Learn more about this topic:

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The Resultant Amplitude of Two Superposed Waves

from MEGA Physics: Practice & Study Guide

Chapter 15 / Lesson 10
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