The wave function (x,t)=(0.03m)sin(2.2x-3.5t) is for a harmonic wave on a string. (a) In what direction does this wave travel and what is its speed? (b) Find the wavelength, frequency, and period of this wave. (c) What is the maximum displacement of any point on the string? (d) What is the maximum speed of any point on the string?
it is y(t) not (x,t)
y is a function of time
to stay on the same spot on the wave
2.2 x - 3.5 t
must be constant
that will happen if
x = (3.5/2.2) t
well, what does that look like ?
distance = speed * time :)
so go right at 3.5/2.2 = 1.59 meters/second
at t = 0 and x = 0, y = 0
when is the argument changed by 2pi at x = 0 ?
3.5 t = 2 pi
t = 2 pi/3.5 = T the period in seconds
similarly where is the argument changed by 2 pi at t = 0?
2.2 x = 2 pi
x = 2 pi/2.2 = Lambda the wavelength in meters
sin or cos can never be bigger than one so max y = 0.02 meter
frequency = 1/T in Hz
max speed = max of dy/dt = 0.03 * 3.5
a. the -3.5t indicates it is moving in the +x direction.
consider some time t+epsilon after time t. Consider the peak of the wave, so (2.2x-3.5t) is 90 degrees. at time t+e, 2.2x-3.5t-3.5e has to be 90 deg, so x has to increase to keep the max.
b.,c: http://www.animations.physics.unsw.edu.au/jw/travelling_sine_wave.htm
To answer your questions step-by-step:
(a) To determine the direction of wave propagation, we need to examine the coefficient of t in the wave function. In this case, the coefficient is -3.5t. Since the coefficient is negative, the wave is traveling in the positive x-direction.
The speed of the wave can be determined by examining the coefficient of x in the wave function. In this case, the coefficient is 2.2. Therefore, the speed of the wave is given by the absolute value of this coefficient, which is 2.2 m/s.
(b) The wavelength of the wave can be found by examining the coefficient of x in the wave function. In this case, the coefficient is 2.2. Therefore, the wavelength of the wave is given by λ = 2π/2.2 ≈ 2.85 m.
The frequency of the wave can be determined using the formula f = v/λ, where v is the speed of the wave. Plugging in the values, we get f = 2.2/2.85 ≈ 0.77 Hz.
The period of the wave can be calculated using the formula T = 1/f, where f is the frequency. Plugging in the value, we get T = 1/0.77 ≈ 1.3 s.
(c) The maximum displacement of any point on the string can be determined by examining the coefficient of sin in the wave function. In this case, the coefficient is 0.03 m. Therefore, the maximum displacement is 0.03 m.
(d) The maximum speed of any point on the string can be determined by taking the derivative of the wave function with respect to time (t). The derivative of sin(2.2x - 3.5t) with respect to t is -3.5(0.03)m cos(2.2x - 3.5t). The maximum value of the cosine function is 1. Therefore, the maximum speed of any point on the string is 3.5(0.03)m ≈ 0.105 m/s.
To summarize:
(a) The wave travels in the positive x-direction.
The speed of the wave is 2.2 m/s.
(b) The wavelength is approximately 2.85 m.
The frequency is approximately 0.77 Hz.
The period is approximately 1.3 s.
(c) The maximum displacement of any point on the string is 0.03 m.
(d) The maximum speed of any point on the string is approximately 0.105 m/s.
To answer these questions, we will analyze the given wave function step by step. Let's address each question individually:
(a) In what direction does this wave travel and what is its speed?
The direction of wave propagation can be determined by examining the coefficient in front of the variable "t" in the wave function. In this case, the coefficient is -3.5t. Since the coefficient is negative, the wave is traveling in the positive x-direction.
Now, let's calculate the speed of the wave. The speed of a wave can be determined by the formula: v = λ * f, where v is the wave speed, λ is the wavelength, and f is the frequency.
(b) Find the wavelength, frequency, and period of this wave.
The wavelength (λ) of the wave is determined by the coefficient in front of the variable "x" in the wave function. In this case, the coefficient is 2.2. Therefore, λ = 2π / 2.2.
The frequency (f) of the wave can be obtained using the equation: f = ω / 2π, where ω is the angular frequency. The angular frequency is the coefficient of "t" multiplied by 2π. So, in this case, ω = -3.5 * 2π.
The period (T) of the wave is the reciprocal of the frequency: T = 1 / f.
(c) What is the maximum displacement of any point on the string?
To find the maximum displacement, we need to determine the maximum value of the sine function. The magnitude of the maximum displacement is equal to the coefficient in front of the sine function. In this case, the coefficient is 0.03 meters.
(d) What is the maximum speed of any point on the string?
The maximum speed occurs when the displacement is at its maximum value. Since the speed of a particle executing simple harmonic motion is given by the product of the angular frequency and the amplitude, the maximum speed is equal to the angular frequency (ω) multiplied by the maximum displacement.
By plugging in the necessary values in the formulas mentioned above, you should be able to obtain the answers to all the questions.