The transverse displacement of a stretched string from equilibrium as a function

Question

The transverse displacement of a stretched string from equilibrium as a function of time and position is given by:
y=0.13 cos(9 x + 54 t). x and y are in m; t is in s.

Part 1--
ANSWERS ARE ONE OF THE FOLLOWING: True, False, Greater Than, Equal To, Less Than
a)The speed of the wave is ... 7 m/s
b) The period is ... 0.1 seconds
c) The wave travels in the negative x direction
d) The wavelength is ... 1 m
HINT: A traveling wave can be any function of (2*pi*x/lamda-2*pi*t/period). Calculate the various parameters where needed then select the proper answers.

Part 2--
Calculate the average power transmitted by the string. Data: mass of a 165 m long piece of the string is 1.87 kg

Explanation / Answer

Here ,

y = A * cos(k * x + w*t)

y = 0.13 * cos(9*x + 54 * t )

w = 54 rad/s

k = 9 rad/s

as wavelength = 2pi/k

wavelength = 2pi/9

wavelength = 0.698 m

frequency = w/2pi

frequency = 54/(2*pi)

frequency = 8.6 Hz

speed of wave = wavelength * frequency

speed of wave = 8.6 * 0.698

speed of wave = 6.003 m/s

period = 1/frequency

period = 1/8.6

period = 0.116 s

hence ,

a)The speed of the wave is LESS THAN 7 m/s
b) The period is GREATER THAN 0.1 seconds
c) The wave travels in the negative x direction TRUE
d) The wavelength is LESS THAN 1 m

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