A wave on a string has a wave function given by y(x, t) = (0.0210 m)sin[(6.91 m^−1)x + (2.38 s^−1)t
.
(a) What is the amplitude of the wave?
(b) What is the period of the wave?
(c) What is the wavelength of the wave?
(d) What is the speed of the wave?
The equation of the wave is
y(x,t)= A•sin(2πx/λ - 2πt/T).
Given equation is
y(x, t) = (0.0210 m)sin[(6.91 m^−1)x + (2.38 s^−1)t
Therefore,
A = 0.0210 m
2π/λ =6.91
λ=2π/6.91=0.909 m
2πt/T= 2.38
T=2πt/ 2.38=2.64 s
λ=vT
v= λ/T=(2π/6.91)/( 2πt/ 2.38)=
=2.38/6.91=0.344 m/s
(a) The amplitude of a wave is the maximum displacement from the equilibrium position. In this case, the wave function is given as follows:
y(x, t) = (0.0210 m) * sin[(6.91 m^−1)x + (2.38 s^−1)t]
Therefore, the amplitude of the wave is 0.0210 m.
(b) The period of a wave is the time it takes for one complete cycle. To find the period, we need to determine the frequency of the wave. The frequency can be obtained from the wave function:
y(x, t) = (0.0210 m) * sin[(6.91 m^−1)x + (2.38 s^−1)t]
The frequency is given by the coefficient of t in the argument of the sine function, which is 2.38 s^−1. The period (T) is then the reciprocal of the frequency:
T = 1 / f = 1 / 2.38 s^−1 = 0.420 s.
Therefore, the period of the wave is 0.420 s.
(c) The wavelength of a wave is the distance over which one complete cycle occurs. It can be determined by examining the coefficient of x in the argument of the sine function. In this case, the coefficient is 6.91 m^−1. The reciprocal of this coefficient gives us the wavelength (λ):
λ = 1 / (6.91 m^−1) = 0.145 m.
Therefore, the wavelength of the wave is 0.145 m.
(d) The speed of a wave can be calculated using the formula:
v = λ * f
where v is the wave speed, λ is the wavelength, and f is the frequency. We have already determined the wavelength (0.145 m) and the frequency (2.38 s^−1). Substituting these values into the formula, we find:
v = (0.145 m) * (2.38 s^−1) = 0.345 m/s.
Therefore, the speed of the wave is 0.345 m/s.
To find the answers to the given questions, we can use the equation for a wave function:
y(x, t) = A * sin(kx + ωt)
where A is the amplitude, k is the wave number, x is the position, ω is the angular frequency, and t is the time.
(a) The amplitude of the wave is given by the coefficient in front of the sine function. In this case, the amplitude is 0.0210 m.
(b) The period of the wave can be determined using the formula:
T = 2π / ω
where T is the period and ω is the angular frequency. In this case, the angular frequency is 2.38 s^(-1). Plugging this into the formula, we get:
T = 2π / 2.38 s^(-1)
Simplifying, we find:
T ≈ 2.64 seconds
(c) The wavelength can be determined from the wave number using the formula:
λ = 2π / k
where λ is the wavelength and k is the wave number. In this case, the wave number is 6.91 m^(-1). Plugging this into the formula, we get:
λ = 2π / 6.91 m^(-1)
Simplifying, we find:
λ ≈ 0.908 meters
(d) The relationship between the speed of a wave, v, the wavelength, λ, and the frequency, f, is given by the formula:
v = λf
Since the period, T, is the inverse of the frequency, we can rewrite the formula as:
v = λ / T
Using the values we've already found, we can calculate the speed:
v ≈ 0.908 meters / 2.64 seconds
Simplifying, we find:
v ≈ 0.344 meters/second