A wave traveling along the x axis is described mathematically by the equation y = 0.15 sin (7.5πt - 0.49πx), where y is the displacement (in meters), t is in seconds, and x is in meters. What is the speed of the wave?
To find the speed of the wave, we need to determine the relationship between the wave's speed, frequency, and wavelength.
In the given equation, y = 0.15 sin (7.5πt - 0.49πx), we can observe that the argument of the sine function is (7.5πt - 0.49πx). This argument represents the phase of the wave.
The general form of a sinusoidal wave traveling along the x-axis can be written as y = A sin (kx - ωt), where A is the amplitude, k is the wave number, ω is the angular frequency, x is the position coordinate, and t is the time coordinate.
Comparing the given equation with the general form, we can extract the values of k and ω as follows:
k = 0.49π
ω = 7.5π
The wave number (k) represents the number of complete wave cycles in a distance of 2π. Therefore, the wavelength (λ) of the wave can be determined as:
λ = 2π/k
Substituting the value of k, we find:
λ = 2π/(0.49π) = 2/0.49 ≈ 4.08 meters
Now, the speed (v) of a wave is given by:
v = λf
where f is the frequency of the wave. The frequency can be obtained from the angular frequency:
f = ω/(2π)
Substituting the value of ω, we obtain:
f = (7.5π)/(2π) = 7.5/2 ≈ 3.75 Hz
Finally, substituting the values of λ and f, we can calculate the speed (v) of the wave:
v = (4.08 meters)(3.75 Hz) ≈ 15.3 meters/second
Therefore, the speed of the wave is approximately 15.3 meters/second.
The wave speed can be determined by examining the equation. In general, the equation for a wave's velocity is given by v = λf, where v is the wave speed, λ is the wavelength, and f is the frequency.
In this case, we can see that the equation for the wave is y = 0.15 sin (7.5πt - 0.49πx).
Next, we need to identify the variables that represent the wavelength and the frequency. The equation can be rewritten as y = 0.15 sin (7.5πt - (0.49/λ)πx), where λ is the wavelength.
Comparing the equation to the general form y = A sin (kx - ωt), we find that k = 2π/λ and ω = 2πf.
In this case, k = 0.49π/λ and ω = 7.5π.
The wave speed v can be calculated as v = ω/k.
Substituting the values we found, we have:
v = (7.5π) / (0.49π/λ)
Simplifying the expression, we get:
v = (7.5π) * (λ / 0.49π)
Next, we can cancel out the π terms:
v = (7.5 * λ) / 0.49
Therefore, the speed of the wave is given by v = (7.5 * λ) / 0.49.