Example 16.2 A Traveling Sinusoidal Wave A sinusoidal wave of wavelength = 40.0

Question

Example 16.2 A Traveling Sinusoidal Wave A sinusoidal wave of wavelength = 40.0 cm and amplitude A = 15.0 cm. The wave function can be written in the form y = A cos(kx t) A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 7.75 Hz. The vertical position of an element of the medium at t = 0 and x = 0 is also 15.0 cm as shown in the figure. (A) Find the wave number k, period T, angular frequency , and speed v of the wave. (B) Determine the phase constant and write a general expression for the wave function. SOLVE IT (A) Find the wave number k, period T, angular frequency , and speed v of the wave. Conceptualize The figure shows the wave at t = 0. Imagine this wave moving to the right and maintaining its shape. Categorize From the description in the problem statement, we see that we are analyzing a mechanical wave moving through a medium, so we categorize the problem with the traveling wave model. Analyze Evaluate the wave number from the equation: k = 2 = 2 rad 40.0 cm = rad/m Evaluate the period of the wave from the equation: T = 1 f = 1 7.75 s1 = s Evaluate the angular frequency of the wave from the equation: = 2f = 2(7.75 s1) = rad/s Evaluate the wave speed from the equation: v = f = (40.0 cm)(7.75 s1) = m/s (B) Determine the phase constant and write a general expression for the wave function. Substitute A = 15.0 cm, y = 15.0 cm, x = 0, and t = 0 into the equation: 15.0 = (15.0)sin sin = 1 = 2 rad Write the wave function: y = A sin kx t + 2 = A cos(kx t) Substitute the values for A, k, and in SI units into this expression. Use the following as necessary: x and t. y = MASTER IT HINTS: GETTING STARTED | I'M STUCK! Another wave shown below at t = 0, is also traveling in the positive x direction. It is given that a = 1.2 cm, b = 88.0 cm, and a frequency of 12.0 Hz. When written in the standard form y = A sin(kx t + ), what is its wave number (in rad/cm), angular frequency, and the phase angle (with 180° < 180° in degrees). wave number rad/cm angular frequency rad/s phase angle ° Sine wave on a coordinate axis, passing through (0,0) with negative slope, of amplitude a and wavelength b.

Explanation / Answer

(A)

k 2p/ = 2pi/40*10^-2 m = 15.7 rad/m

T = 1/f = 1/7.75s-1 = 0.129 s

w =2pf = 6.28rad(7.75s-1) = 48.7 rad/s

v = f = 40*10^-2 m (7.75s-1) = 3.10 m/s

(B)

At x=0 and t=0, y=15.0cm, therefore the phase becomes

y = 0.15 * sin = 0.15

Here = / 2, so sin = 1

Thus the general wave function is

y = A sin( kx t + )

y = 0.15 sin (15.7x 48.7t + / 2 )

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