) A cord has two sections with linear densities of and Fig. 15–34. An incident wave, given by where x is in meters and in seconds, travels along the lighter cord. (a) What is the wavelength on the lighter section of the cord? (b) What is the tension in the cord? (c) What is the wavelength when the wave travels on the heavier section?
In order to answer the questions, we need to use some basic formulas related to waves, linear density, and tension.
(a) To find the wavelength on the lighter section of the cord, we can use the formula:
wavelength = velocity / frequency
The linear density is given by λ1 = m1 / L1, where λ1 is the linear density of the lighter section, m1 is the mass of the section, and L1 is the length of the section. In Fig. 15–34, the lighter section of the cord is denoted by subscript 1.
We are also given the incident wave equation y(x,t) = A cos(kx - ωt), where A is the amplitude, k is the wave number, x is the position along the cord, ω is the angular frequency, and t is the time.
Since the wave travels along the lighter cord, we can write the equation as y(x,t) = A1 cos(k1x - ωt), where A1 is the amplitude on the lighter section, and k1 is the wave number on the lighter section.
The wave number is related to the wave speed (v) and the angular frequency (ω) by the formula k = ω / v.
Since the linear density λ1 is related to the wave speed v by the formula v = √(T/λ1), where T is the tension in the cord, we can rearrange it to find λ1 = T / (v^2).
Combining these equations, we get:
k1 = ω / v1 = ω √(λ1 / T)
Since the wave number k is related to the wavelength λ by the formula k = 2π / λ, we can rearrange it to find the wavelength:
λ1 = 2π / k1 = 2π / (ω √(λ1 / T))
To solve this equation, we need the value of angular frequency (ω). If it is not given in the question, we'll need to assume a value or find it from additional information.
(b) To find the tension in the cord, we can use the formula:
T = μv^2
where μ is the linear mass density, and v is the wave speed.
In this case, since we are given the linear mass density λ1 for the lighter section, we can rearrange the formula to find the tension:
T = λ1v^2
We also need the value of the wave speed (v) to calculate the tension. If it is not given in the question, we'll need to assume a value or find it from additional information.
(c) To find the wavelength when the wave travels on the heavier section of the cord, we can use similar concepts and formulas as in part (a), but with the appropriate values for the linear density (λ2) and tension (T2) of the heavier section of the cord. We need to substitute λ2 and T2 in the equation:
λ2 = 2π / (ω √(λ2 / T2))
Again, we'll need the value of the angular frequency (ω) to solve this equation. If it is not given, we'll need to assume a value or find it from additional information.