PX154 - H2 - two strings joined together - transmission and reflection

introduction

• string 1, and string 2 are joined at $x_0$
• the join is perfect
• mass per unit length of string 1 is ${\mu }_1$, and that for string 2 is ${\mu }_2$
• tension, $T$, and wave frequency, $\omega$, are same in both strings
• wave speed, $v=\sqrt{\frac{T}{\mu }}$ , is slower in the heavier string
• the frequency, $f$, should be the same in both the strings, otherwise the waves will go out of phase and the strings will come apart
• the wavelength, $\lambda =\frac{v}{f}$, is shorter in the heavier string as $v$ is slower, and $f$ is constant

application

• concepts apply elsewhere:

  • light from one medium into another, eg: air to glass
  • QM treatment of particles, eg: at potential step
  • electrical signals in electronics

• in general, expect three waves:

  • incident wave:
  $$y_i(x,t)=A_i\cos (k_{1}x-\omega t)$$
  • reflected wave:
  $$y_r(x,t)=A_r\cos (k_{1}x-\omega t)$$
  • transmitted wave:
  $$y_t(x,t)=A_t\cos (k_{2}x-\omega t)$$

what happens at the join?

• boundary conditions:
  • string is continuous (doesn't snap):
  $$y_{string1}(x_0)=y_{string2}(x_0)⟹y_i+y_r=y_t$$
  • tension is the same in both strings
    • at the joint, the forces are equal
    • string only moves in the transition
• consider PX154 - G1a - waves on a taut string for forces on an element of the string
  
• no change in gradient at the joint
• at $x_0$:

• using these conditions, the reflection and transmission coefficients, $r$ and $t$ can be derived:

where, $z=\sqrt{\mu T}$ is the impedance

observation

• for a wave approaching the join:
• if ${\mu }_1<{\mu }_2$:
• if ${\mu }_1>{\mu }_2$:
• if ${\mu }_1={\mu }_2$:

power transmission at the boundary

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