To determine the properties of the traveling wave described by the equation y(x, t) = 0.004 * cos(20x + 200t), we can examine the coefficients within the equation. Let's break it down:
The general form of a traveling wave is given by y(x, t) = A * cos(kx - ωt + φ),
where: A is the amplitude, k is the wave number, ω is the angular frequency, φ is the phase constant.
Comparing this to the given equation, we can deduce the properties of the wave:
Amplitude (A): The amplitude is the coefficient of the cosine term. In this case, A = 0.004 meters.
Angular frequency (ω): The angular frequency is the coefficient of 't' within the cosine term. Here, ω = 200 rad/s.
Wave number (k): The wave number can be obtained by comparing the coefficient of 'x' within the cosine term. In this case, k = 20 rad/m.
Frequency (f): The frequency is related to the angular frequency by the equation f = ω / (2π). Substituting the given value, we find f = 200 / (2π) ≈ 31.83 Hz.
Wavelength (λ): The wavelength is the distance between corresponding points on the wave. It can be determined using the formula λ = 2π / k. Plugging in the given value of k, we get λ = 2π / 20 ≈ 0.314 meters.
Speed of the wave (v): The speed of the wave can be calculated by multiplying the frequency (f) by the wavelength (λ). Thus, v = f * λ ≈ 31.83 Hz * 0.314 m ≈ 9.99 m/s.
Direction of travel: The direction of travel can be inferred from the wave number (k). In this case, since k is positive, the wave is traveling in the positive x-direction.
To summarize:
- Amplitude: 0.004 meters
- Frequency: 31.83 Hz
- Wavelength: 0.314 meters
- Speed: 9.99 m/s
- Direction of travel: Positive x-direction