A wave on a string is described by the relation y = A sin(30t - 0.041x), where t is measured in seconds and x in meters, with A = 0.16 m.
(a)Find the frequency of the wave. ____Hz
(b)Find the wavelength of the wave. ___m
(c)Find the speed of the wave. ___m/s
y = A sin(30t - 0.041x)
y = -Asin(0.041x - 30t)
this is of the form y = A*sin(k*x - omega*t)
k = 0.041
omega = 30
where omega = 2*pi*f; 30 = 2*pi*f; where f is frequency; use this to solve for answer in a)
b) k = 2*pi / lambda, where lambda is wavelength;
c) the speed of the wave is omega/k = lambda*f
To find the frequency of the wave, we need to determine the coefficient of the time variable t in the given equation. The coefficient in front of the t term represents the angular frequency (ω) of the wave, which is related to the frequency (f) by the equation:
f = ω/2π
In the given equation, the coefficient of t is 30.
Therefore, the angular frequency ω = 30.
Substituting this value into the equation for frequency f = ω/2π:
f = 30/2π ≈ 4.7746 Hz
Therefore, the frequency of the wave is approximately 4.7746 Hz.
To find the wavelength of the wave, we need to determine the coefficient of the space variable x in the given equation. The coefficient in front of the x term represents the angular wave number (k), which is related to the wavelength (λ) by the equation:
k = 2π/λ
In the given equation, the coefficient of x is -0.041.
Therefore, the angular wave number k = -0.041.
Substituting this value into the equation for wave number k = 2π/λ:
-0.041 = 2π/λ
Rearranging the equation to solve for λ:
λ = 2π/(-0.041) ≈ -153.9269 m
Note that the wavelength is a positive value, so we can take its absolute value:
λ ≈ 153.9269 m
Therefore, the wavelength of the wave is approximately 153.9269 m.
To find the speed of the wave, we can use the relationship between the frequency (f) and wavelength (λ):
v = fλ
Substituting the values we found:
v = (4.7746 Hz)(153.9269 m) ≈ 734.4709 m/s
Therefore, the speed of the wave is approximately 734.4709 m/s.