A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m−1 x). What is the velocity of the wave?
Question
A transverse periodic wave is represented by the equation y(x, t) = 2.50 cm cos(2,500 rad/s t − 15.0 m−1 x). What is the velocity of the wave?
Solution
The velocity of a wave can be determined using the wave equation, which is given by:
v = fλ
where: v is the velocity, f is the frequency, and λ is the wavelength.
However, in this case, we are given the wave equation in the form:
y(x, t) = A cos(ωt - kx)
where: A is the amplitude, ω is the angular frequency, k is the wave number, and x is the position.
From this equation, we can see that the wave number k is 15.0 m^-1 and the angular frequency ω is 2,500 rad/s.
The wave number k is related to the wavelength λ by the equation:
k = 2π/λ
Rearranging for λ gives:
λ = 2π/k
Substituting the given value for k gives:
λ = 2π/15.0 m^-1 = 0.42 m
The angular frequency ω is related to the frequency f by the equation:
ω = 2πf
Rearranging for f gives:
f = ω/2π
Substituting the given value for ω gives:
f = 2,500 rad/s / 2π = 398 Hz
Now we can substitute the values for f and λ into the wave equation to find the velocity v:
v = fλ = 398 Hz * 0.42 m = 167 m/s
So, the velocity of the wave is 167 m/s.
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