Two waves are combined to form this standing wave equation of
y(x,t) = (3 mm)sin[x/(2 m]cos[(100 rad/s)t]
a. What's the amplitude of the right moving wave and left moving wave?
b. What's the wavelength of the right moving wave and left moving wave?
c. What's the velocity of the right moving wave and the left moving wave?
a)
when they add up, the maximum will be twice the amplitude of the individual right and left waves so 1.5 mm each
b)
a wavelength is when x/2 = 2 pi
or x = 4 pi
c) 2 pi f = 100 and lambda = 4 pi
T = 2 pi/100 = pi/50
distance = rate * time
4 pi = speed * pi/50
speed = 50/4 = 12.5 m/s
one is +12.5 and the other is -12.5
Thanks.
To answer these questions, we need to understand the components of the standing wave equation you provided.
The given standing wave equation is y(x, t) = (3 mm)sin[x/(2 m)]cos[(100 rad/s)t].
a. The amplitude of a wave represents the maximum displacement from the equilibrium position. In the given equation, the amplitude for both the right and left moving waves is 3 mm. This is because the term "3 mm" appears right at the beginning of the equation, outside the sine and cosine functions, and it determines the overall amplitude.
b. The wavelength of a wave represents the distance between two adjacent peaks (or troughs) of the wave. To find the wavelength of the right and left moving waves, we need to separate the equation into components.
For the right-moving wave, we only consider the cosine function, so we ignore the sine function. In this case, the wavelength is given by λ = 2π/k, where k is the wave number, which is determined by the argument of the cosine function. In our equation, the argument is (100 rad/s)t. Since the coefficient of t is 100 rad/s, we can use the relationship ω = 2πf = 100 rad/s, where ω is the angular frequency and f is the frequency. Thus, the angular frequency is ω = 100 rad/s.
By comparing ω with the general equation ω = ck, where c is the wave speed and k is the wave number, we find that the wave speed c equals 100 rad/s / k. However, we are given the wave number, but not the wave speed, so we cannot directly calculate the wavelength of the right-moving wave without additional information.
Similarly, for the left-moving wave, we only need to consider the sine function. Again, we do not have enough information to calculate the wavelength of the left-moving wave.
c. The velocity of a wave is given by v = λf, where λ is the wavelength and f is the frequency. Since we couldn't determine the wavelength for either the right or left-moving wave, we also cannot calculate their velocities.