A cord has two sections with linear densities of 0.10 kg/m and 0.20 kg/m respectively. An incident wave, y(x,t) = 0.050sin(6.0x - 12.0t), where x is in meters and t in seconds, travels from the lighter cord to the heavier one. What is the wavelength of the wave in each section of the cord?
So I interpret the wave going through like this:
|--------1--------|--------2--------|
In cord 1, the linear density is u_1 = 0.10kg/m. In cord 2, the density u_2 = 0.20kg/m.
The wave goes through cord 1 first, and then through cord 2.
Using the template y(x,t) = Asin(kx - wt) and from the given function y(x,t) = 0.050sin(6.0x - 12.0t):
k = 6.0 m^-1
w = 12.0 rad/s
k = 2pi/lambda -> lambda = 2pi/k = (2)(pi)/6.0m^-1 = 1.047197551m = 1.05m
And I guess that lambda_1 = 1.05m then?
But how do I get lambda_2 (the wavelength of the wave travelling through cord 2)?
The frequency is the same on both sides of the transition. There is no box to store waves in so as many have to pass a point in an hour on the left as on the right. Same f and same w and same 1/f = period T
The tension is the same both sides
w = 2 pi f = the same both sides
the s
in the light line
12/f = 2 pi
so f = 6/pi and w = 2 pi f = 12rad/s agreed
for wavelength
6.0 lambda = 2 pi
lambda = pi/3 = 1.05 sure enough
NOW
Double the mass per unit length
speed of transverse wave = sqrt(tension/mass per unit length)
so
the speed is the original speed /sqrt 2
so in the same amount of time (period) it goes original lambda/sqrt 2
= 1.05/sqrt 2
How did you get "sqrt 2" from "sqrt(tension/mass per unit length)"?
Well, since you're dealing with a wave traveling from the lighter cord to the heavier one, you can use the concept of impedance to find the wavelength in each section.
The impedance of a medium is given by the product of its linear density and the square of the wave speed in that medium. In this case, the wave speed depends on the tension in the cord and the linear density.
Let's call the tension in cord 1 T_1 and the tension in cord 2 T_2.
The wave speed in cord 1 is given by:
v_1 = sqrt(T_1 / u_1)
And the wave speed in cord 2 is given by:
v_2 = sqrt(T_2 / u_2)
Since the wave is continuous, the wave speed must be the same in both cords. Therefore, we have:
v_1 = v_2
Now, we can use the formula for the wave speed to express the tensions in terms of the linear densities and wave speeds:
T_1 = u_1 * v_1^2
T_2 = u_2 * v_2^2
We know that the frequency of the wave remains the same as it travels from cord 1 to cord 2. So we can use the formula for the wavelength of a wave to find lambda_2:
lambda_2 = v_2 / f = v_1 / f = lambda_1
Therefore, the wavelength of the wave in cord 2 is the same as the wavelength in cord 1.
So, lambda_2 = 1.05m, just like lambda_1.
And now equations are all tangled up, just like headphone wires in the depths of your bag!
To find the wavelength of the wave traveling through cord 2, you can use the principle of continuity. The principle of continuity states that when a wave passes from one medium to another, the frequency remains constant, but the velocity and wavelength may change.
Since the linear density of cord 2 is different from cord 1, the velocity of the wave will be different in the two sections. The velocity of a wave on a string is given by the equation v = sqrt(T/u), where T is the tension in the string and u is the linear density of the string.
Let's assume the tension in the cord remains constant throughout. Therefore, the velocity of wave 1 in cord 1 is v_1 = sqrt(T/u_1), and the velocity of wave 2 in cord 2 is v_2 = sqrt(T/u_2).
Since the frequency remains constant, we can use the equation v = lambda * f, where v is the velocity, lambda is the wavelength, and f is the frequency.
In cord 1, we have v_1 = lambda_1 * f, and in cord 2, we have v_2 = lambda_2 * f.
Since the frequency is constant, we can equate the two expressions for velocity in terms of wavelength:
v_1 = v_2
lambda_1 * f = lambda_2 * f
Canceling out the frequency, we find that:
lambda_1 = (v_1/v_2) * lambda_2
Substituting the expressions for v_1 and v_2, we have:
lambda_1 = (sqrt(T/u_1)) / (sqrt(T/u_2)) * lambda_2
Simplifying further:
lambda_1 = sqrt(u_2/u_1) * lambda_2
Now, we can substitute the given values into the equation to find lambda_2:
lambda_1 = sqrt(0.20 kg/m / 0.10 kg/m) * 1.05 m
lambda_1 = sqrt(2) * 1.05 m
lambda_1 ≈ 1.48 m
So, the wavelength of the wave in cord 2 is approximately 1.48 meters.