15.44 • The wave function of a standing wave is y(x, t) = 4.44 mm * sin[(32.5rad / m) * x] * sin[(754rad / s) * t] . For the two travel ing waves that make up this standing wave, find the (a) amplitude; (b) wavelength; (c) frequency; (d) wave speed; (e) wave functions. (f) From the information given, can you determine which har monic this is? Explain.
To find the properties of the traveling waves that make up the standing wave, we can analyze the given wave function.
The general form of a standing wave is y(x, t) = 2A * sin(kx) * sin(ωt), where A is the amplitude, k is the wave number, ω is the angular frequency, and x and t represent the position and time variables, respectively.
Comparing this general form with the given wave function, we can identify the properties:
(a) Amplitude (A): In the given wave function, the amplitude is given as 4.44 mm.
(b) Wave number (k): The wave number is related to the wavelength (λ) by the equation k = 2π / λ. Comparing this with the given wave function, we can see that the wave number is 32.5 rad/m. To find the wavelength, we can rearrange the equation to λ = 2π / k.
λ = 2π / (32.5 rad/m) ≈ 0.194 m
Therefore, the wavelength is approximately 0.194 m.
(c) Frequency (f): The frequency is related to the angular frequency (ω) by the equation ω = 2πf. Comparing this with the given wave function, we can see that the angular frequency is 754 rad/s. To find the frequency, we can rearrange the equation to f = ω / (2π).
f = (754 rad/s) / (2π) ≈ 120 Hz
Therefore, the frequency is approximately 120 Hz.
(d) Wave speed (v): The wave speed is given by the equation v = fλ. By substituting the previously determined values for frequency and wavelength, we can calculate the wave speed.
v = (120 Hz) * (0.194 m) ≈ 23.28 m/s
Therefore, the wave speed is approximately 23.28 m/s.
(e) Wave functions: The given wave function already represents the two traveling waves that make up the standing wave. It can be written as follows:
y(x, t) = 4.44 mm * sin[(32.5 rad / m) * x] * sin[(754 rad / s) * t]
(f) Harmonic determination: From the information given, we cannot determine which harmonic this is. The harmonic number refers to the number of half-wavelengths present in the standing wave. To determine the harmonic, we need additional information such as the length of the medium or the number of nodes and antinodes.
In summary:
(a) Amplitude: 4.44 mm
(b) Wavelength: Approximately 0.194 m
(c) Frequency: Approximately 120 Hz
(d) Wave speed: Approximately 23.28 m/s
(e) Wave functions: y(x, t) = 4.44 mm * sin[(32.5 rad / m) * x] * sin[(754 rad / s) * t]
(f) Harmonic determination: Not enough information provided