Wave Propagation in Connected Strings

Understanding reflection and transmission at density boundaries

Consider a system of three connected strings, \( S_1 \), \( S_2 \) and \( S_3 \) with uniform linear mass densities \( \mu \) kg/m, \( 4\mu \) kg/m and \( 16\mu \) kg/m, respectively, as shown in the figure. \( S_1 \) and \( S_2 \) are connected at the point \( P \), whereas \( S_2 \) and \( S_3 \) are connected at the point \( Q \), and the other end of \( S_3 \) is connected to a wall. A wave generator \( O \) is connected to the free end of \( S_1 \). The wave from the generator is represented by \( y = y_0 \cos(\omega t - kx) \) cm, where \( y_0 \), \( \omega \) and \( k \) are constants of appropriate dimensions. Which of the following statements is/are correct:

Wave Generator (O) ───[S₁:μ]───(P)───[S₂:4μ]───(Q)───[S₃:16μ]─── Wall

(A) When the wave reflects from \( P \) for the first time, the reflected wave is represented by \( y = a_1 y_0 \cos(\omega t + kx + \pi) \) cm, where \( a_1 \) is a positive constant.

(B) When the wave transmits through \( P \) for the first time, the transmitted wave is represented by \( y = a_2 y_0 \cos(\omega t - kx) \) cm, where \( a_2 \) is a positive constant.

(C) When the wave reflects from \( Q \) for the first time, the reflected wave is represented by \( y = a_3 y_0 \cos(\omega t - kx + \pi) \) cm, where \( a_3 \) is a positive constant.

(D) When the wave transmits through \( Q \) for the first time, the transmitted wave is represented by \( y = a_4 y_0 \cos(\omega t - 4kx) \) cm, where \( a_4 \) is a positive constant.

Solution

The correct answers are A and D.

Explanation:

About point P (μ to 4μ interface):

1. Reflection coefficient \( r_1 = \frac{Z_1 - Z_2}{Z_1 + Z_2} = \frac{\sqrt{\mu} - \sqrt{4\mu}}{\sqrt{\mu} + \sqrt{4\mu}} = \frac{1-2}{1+2} = -\frac{1}{3} \)

2. Transmission coefficient \( t_1 = \frac{2Z_1}{Z_1 + Z_2} = \frac{2\sqrt{\mu}}{\sqrt{\mu} + 2\sqrt{\mu}} = \frac{2}{3} \)

3. The reflected wave has amplitude \( a_1 = \frac{1}{3} \) and phase change of π (due to negative coefficient)

4. The transmitted wave has amplitude \( a_2 = \frac{2}{3} \) but wave number changes to \( k' = \frac{k}{2} \) (since \( v = \sqrt{T/μ} \) and \( ω = vk \) remains constant)

About point Q (4μ to 16μ interface):

1. Reflection coefficient \( r_2 = \frac{Z_2 - Z_3}{Z_2 + Z_3} = \frac{2-4}{2+4} = -\frac{1}{3} \)

2. Transmission coefficient \( t_2 = \frac{2Z_2}{Z_2 + Z_3} = \frac{4}{6} = \frac{2}{3} \)

3. The reflected wave has amplitude \( a_3 = \frac{1}{3} \) and phase change of π

4. The transmitted wave has amplitude \( a_4 = \frac{2}{3} \) but wave number changes to \( k'' = \frac{k}{4} \) (which is equivalent to 4kx in the denominator)

Therefore:

- A is correct: Reflection at P has phase change π

- B is incorrect: Transmitted wave should have different k value

- C is incorrect: Reflected wave from Q should have +kx (travels backward)

- D is correct: Transmitted wave through Q has k reduced by factor of 4

Wave Propagation Visualization

String S₁

Density: μ kg/m

Impedance: Z₁ = √μ

String S₂

Density: 4μ kg/m

Impedance: Z₂ = 2√μ

String S₃

Density: 16μ kg/m

Impedance: Z₃ = 4√μ

Note: This is a simplified representation showing wave behavior at boundaries.

💡 Did you know? The principle of wave reflection and transmission at impedance boundaries explains why some musical instruments produce richer harmonics than others. The careful matching of string densities and tensions allows for optimal energy transfer and standing wave formation!