A wave traveling in the +x direction has an amplitude of 0.45 m, a speed of 6.1 m/s, and a frequency of 16 Hz. Write the equation of the wave in the form given by either Equation 16.3 or 16.4. (Answer in terms of t and x. Assume standard units.)
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we are in the same class
To write the equation of the wave in the given form, we need to use the wave equation:
y(x, t) = A * sin(kx - ωt + φ)
Where:
- y(x, t) is the displacement of the wave at position x and time t
- A is the amplitude of the wave
- k is the wave number (2π/λ), where λ is the wavelength
- ω is the angular frequency (2πf), where f is the frequency
- φ is the phase constant
First, let's find the values of k and ω:
Given: A = 0.45 m, v = 6.1 m/s, f = 16 Hz
The wave equation tells us that v = λf, where λ is the wavelength. Rearranging the equation, we have:
λ = v/f
Substituting the given values:
λ = 6.1 m/s / 16 Hz ≈ 0.38125 m
Next, we can find k and ω using the formulas:
k = 2π / λ
ω = 2πf
Substituting the values:
k = 2π / 0.38125 m ≈ 16.516 rad/m
ω = 2π * 16 Hz ≈ 100.531 rad/s
Now we have all the values needed to write the equation. However, we still need to find the phase constant φ.
To find the phase constant φ, we need additional information about the wave, such as its initial position or initial velocity, which is not provided in your question. Without this information, we cannot determine the specific value of φ.
Therefore, we can write the equation of the wave as:
y(x, t) = 0.45 * sin(16.516x - 100.531t + φ)
But the value of φ is unknown without extra information.