A wave traveling along the x axis is described mathematically by the equation y = 0.16 sin (9.0πt - 0.58πx), where y is the displacement (in meters), t is in seconds, and x is in meters. What is the speed of the wave?
Well, calculating the speed of a wave is quite a wave-y subject! But fear not, for Clown Bot is here to make it fun!
To find the speed, we need to determine how fast the wave is moving along the x-axis. The equation given is in the form y = A sin (kx - ωt), where k is the wave number and ω is the angular frequency.
In this case, the wave number is given as 9.0π, and since the wave number is related to the wavelength (λ) and the angular frequency (ω) by the formula k = 2π/λ, we can conclude that the wavelength is λ = (2π) / (9π) = 2 / 9 meters.
The angular frequency ω is the coefficient in front of the t variable, which is 9.0π in this case.
Now, the speed of the wave can be determined by the formula v = λω. Plugging in the values, we get:
v = (2 / 9) meters * (9.0π radians/second)
v ≈ 2π meters/second
So, the speed of the wave is approximately 2π meters/second. And remember, waves are notorious for making a splash!
To find the speed of a wave, we need to determine the relationship between the wave's speed, frequency, and wavelength.
In the given equation, y = 0.16 sin(9.0πt - 0.58πx), we can see that the wave is sine-shaped with the form sin(kx - ωt), where k represents the wave number (2π divided by the wavelength λ), and ω is the angular frequency (equal to 2π times the frequency f).
Comparing the equation to the given equation y = 0.16 sin(9.0πt - 0.58πx), we can deduce that:
k = 0.58π
ω = 9.0π
The wave speed (v) is equal to the product of the angular frequency (ω) and the wavelength (λ).
v = ωλ
Since we know that k = 2π/λ, we can rearrange the equation to solve for λ:
λ = 2π / k
Plugging in the value of k:
λ = 2π / 0.58π
π cancels out:
λ = 2 / 0.58
Simplifying, we find:
λ ≈ 3.45 meters
Now we can calculate the speed (v) by substituting the values of ω and λ into the equation:
v = ωλ
v = (9.0π) × (3.45)
Using a calculator, we find:
v ≈ 97.43 m/s
Therefore, the speed of the wave is approximately 97.43 meters per second.
To find the speed of the wave, we need to determine the relationship between the wave's angular frequency (ω) and the wave number (k).
The equation for a wave traveling along the x-axis is given by the general form:
y(x, t) = A sin(kx - ωt + ϕ)
Comparing this equation with the provided equation: y = 0.16 sin (9.0πt - 0.58πx), we can see that the angular frequency is given by ω = 9.0π and the wave number is given by k = 0.58π.
The speed of the wave can be calculated using the formula:
v = ω / k
Substituting the given values, we have:
v = (9.0π) / (0.58π)
Now, cancel out the common factor of π:
v ≈ 9.0 / 0.58
Evaluating this expression gives:
v ≈ 15.52 m/s
Therefore, the speed of the wave is approximately 15.52 m/s.