Reflection and Transmission of Wave at Boundary of Medium: Wave on a Composite S

Question

Reflection and Transmission of Wave at Boundary of Medium: Wave on a Composite String "medium 1 "medium 2" incident wave "i" directed transmitted wave "t" directed reflected wave "r" directed A string is made up of two parts. The left side has linear mass density and the right side has a greater mass density 2 > . The string is under tension F,ension (must be the same in both parts). Therefore the wave speed will change when a wave reaches the boundary between the two media The following "boundary conditions must be true: (J): The strings are connected, so at any time: yx 0)2x0) (Il): There cannot be a kink in the string, where the two strings meet: at x The wave function for the incident sinusoidal wave is ax ax (xt)-A sin(ot-kx) (wave is traveling to the right) The wave function for the reflected wave is y, (xl)-A, sin(ons + k) (note, this wave is traveling to the left) Both of these wave superpose, so that(,)-y (xt)+y, (x.t) The wave transmitted into medium 2 can be described as: 2(xt)y(x,t)-A, sin (ot -kx)

Explanation / Answer

given

medium 2 has mass density of mu2, medium 1 has mu1

and mu2 > m1

henec, this acts as a hard boundary when the incident wave comes from medium 1 to mediuym 2, and as a soft medium when the wave travels form medium 2 to medium 1

a. Comparing tha angular frequencies

As angular frequency is dependent only on the source of the wave and not on the medium, both the angular fireuqncies, in both media will be the same

so, w1 = w2

b. speed of wave in medium1, v1 = sqroot(T/mu1) [ where T is the common tensino in both the strings]

speed of wave in medium 2, v2 = sqroot(T/mu2) [ where T is the common tensino in both the strings]

now, for a wave with wavenumber k and angular frequency w

v = w/k

but w1 = w2

so, v1k1 = v2k2

sqroot(T/mu1)*k1 = sqroot(T/mu2)*k2

k1^2/mu1 = k2^2/mu2

for, mu2 > mu1

k2 < k1

c. y1(x = 0) = y2 (x = 0)

now y1 = Ai*sin(wt - k1x) + Ar*sin(wt + k1x)

y2 = Atsin(wt - k2x)

applygin boundary conditions

Ai*sin(wt) + Ar*sin(wt) = Atsin(wt)

or Ai + Ar = At

d. dy1/dx at x = 0 = dy2/dx at x = 0

d(Ai*sin(wt - k1x) + Ar*sin(wt + k1x) )/dx = d(Atsin(wt - k2x))/dx at x = 0

Ai*cos(wt - k1x)*(-k1) + Ar*cos(wt + k1x)*k1 = At*cos(wt - k2x)*(-k2)

at x = 0

-k1*Ai + k1*Ar = -k2*At

(Ai - Ar)K1 = k2*At

  

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