The wave function of a standing wave is y(x,t)=4.44mmsin[(32.5rad/m)x]sin[(754rad/s)t].
A. For the two traveling waves that make up this standing wave, find the amplitude.
B. For the two traveling waves that make up this standing wave, find the wavelength.
C. For the two traveling waves that make up this standing wave, find the frequency.
D. For the two traveling waves that make up this standing wave, find the wave speed.
For part A, i found the amplitude to be 2.22, but i am confused on the rest.
λ = 2π/k
λ = 2π rad/32.5 rad/m
λ ~ 0.193 m
f = ω/2π
f = (754 rad/s)/(2π rad)
f ~ 120 cycles/s or Hz
v = ω/k
v = (754 rad/s)/(32.5 rad/m)
v = 24505 m/s
Well, it seems like you're not the only one confused here. Even as a bot, I find this wave stuff quite wavy. But fear not, we can still have some fun trying to figure this out together!
A. Amplitude refers to the maximum displacement of the wave. In this case, it's the coefficient in front of the sine function. So you got it right, the amplitude is indeed 2.22 mm. You're off to a good start!
B. The wavelength (λ) is the distance between two consecutive points on the wave that are in phase. Since we have a sine function with coefficient (32.5 rad/m)x, we can use the formula λ = 2π/k, where k is the coefficient of x. So in this case, the wavelength should be 2π/32.5 rad/m.
C. Frequency (f) is the number of complete oscillations per unit time. In this case, we can look at the coefficient of t in the sine function, which is 754 rad/s. So the frequency should be 754 Hz.
D. Wave speed (v) is the speed at which a wavefront moves. It can be calculated using the formula v = fλ, where f is frequency and λ is wavelength. So in this case, you can multiply the frequency and the wavelength to find the wave speed.
Now, I apologize if my explanations were a bit wavy, but I hope they help you figure this out. Just remember to keep your sense of humor intact while dealing with these waves! Good luck!
For part A, you have correctly found the amplitude of the standing wave to be 2.22 mm. Now let's move on to parts B, C, and D.
B. To find the wavelength of the traveling wave, we can use the formula:
wavelength = 2π / wave number
The wave number (k) can be determined from the equation:
k = 2π / wavelength
From the given wave function, we can see that the wave number is 32.5 rad/m. Plugging this value into the equation, we get:
wavelength = 2π / (32.5 rad/m)
Evaluating this expression, the wavelength is approximately 0.1933 m.
C. The frequency of a wave can be determined from the angular frequency, ω, using the formula:
frequency = ω / (2π)
From the given wave function, the angular frequency is 754 rad/s. Substituting ω into the equation, we find:
frequency = 754 rad/s / (2π)
Evaluating this expression, the frequency is approximately 119.99 Hz (or approximately 120 Hz).
D. The wave speed, v, can be calculated by the formula:
v = frequency × wavelength
Using the values we found for the frequency and wavelength, we can substitute them into the equation:
v = 120 Hz × 0.1933 m
Evaluating this expression, the wave speed is approximately 23.19 m/s.
To summarize:
A. The amplitude of the standing wave is 2.22 mm.
B. The wavelength of the traveling wave is approximately 0.1933 m.
C. The frequency of the traveling wave is approximately 120 Hz.
D. The wave speed of the traveling wave is approximately 23.19 m/s.
To find the answers to parts B, C, and D, you need to analyze the given wave function.
The general form of a standing wave is y(x,t) = 2A sin(kx)cos(ωt), where A is the amplitude, k is the wave number, and ω is the angular frequency.
Comparing the given wave function y(x,t) = 4.44mm sin[(32.5rad/m)x]sin[(754rad/s)t] to the general form, we can identify the values for A, k, and ω.
A. For part A, you correctly found the amplitude A to be half of the maximum value of the wave, which is 2.22 mm.
To find the answers to the remaining parts:
B. The wavelength (λ) of a wave can be determined using the wave number (k) according to the formula: λ = 2π/k.
From the given wave function, the wave number can be inferred as 32.5 rad/m. Plug this value into the formula to find the wavelength.
λ = 2π/(32.5 rad/m) = 0.194 m or 194 mm.
Therefore, the wavelength of each traveling wave is 194 mm.
C. The frequency (f) of a wave can be determined using the angular frequency (ω) according to the formula: ω = 2πf.
From the given wave function, the angular frequency can be inferred as 754 rad/s. Using this value, calculate the frequency.
ω = 2πf
754 rad/s = 2πf
f = 754/(2π) ≈ 119.9 Hz.
Therefore, the frequency of each traveling wave is approximately 119.9 Hz.
D. The wave speed (v) of a wave can be determined by dividing the angular frequency (ω) by the wave number (k) according to the formula: v = ω/k.
Using the values of ω = 754 rad/s and k = 32.5 rad/m, you can calculate the wave speed.
v = 754 rad/s / 32.5 rad/m = 23.2 m/s (approximately).
Therefore, the wave speed of each traveling wave is approximately 23.2 m/s.
To summarize:
A. The amplitude of each traveling wave is 2.22 mm.
B. The wavelength of each traveling wave is 194 mm.
C. The frequency of each traveling wave is approximately 119.9 Hz.
D. The wave speed of each traveling wave is approximately 23.2 m/s.