(10 points) A string of length L/2 and linear mass density is attached to a stri

Question

(10 points) A string of length L/2 and linear mass density is attached to a string of length L/2 and linear mass density 2, as shown below. The left end of the string of mass density and the right end of the string of m ass density 2 are both attached to rigid walls with y(t) = 0. In this problem you will eventually determine an expression for the eigenfrequencies for standing waves 1 y-o x=L/2 2x1 x-0 Assume that the transverse displacement for the left half of the string is given by Y1(x, t) = Ai sin(kix + ) cos(wit), and the transverse displacement for the right half of the string is (a) Does w Defend your answer qualitatively. (b) Use the appropriate boundary conditions at x 0 and x L to determine and constrain (express 2 in terms of constants and K2). (c) What are the appropriate boundary conditions at x = L/2?

Explanation / Answer

given, two strings are attached
left String:
length : L/2
mass density = mu1

Right String
length : L/2
mass density = mu2

assuming transverse displacement of left half of the striing
y1(x,t) = A1sin(k1x + phi1)cos(w1t)
y2(x,t) = A2sin(k2x + phi2)cos(w2t)

a) As w , the angular frequency is dependent on oscillating source and not on medium properties, the w1 = w2 will be true for both the strings when disturbed simultaneously by the same source.
   hence w1 = w2 = w
b) considering the left boundary condition:
   y1(0,t) = 0
   A1sin(phi1)cos(w1t) = 0
   hence phi1 = 0

   similiarly from the right boundary condition
   y2(0,t) = 0
   phi2 = 0

c)   also, at x = L/2
   y1 = y2
   and d(y1)/dx = d(y2/dx)

   A1sin(k1x)cos(wt) = A2sin(k2x)cos(wt)
   A1sin(k1x) = A2sin(k2x) ( at x = L/2)
   A1sin(k1*L/2) = A2*sin(k2*L/2) ... (1)

   also,
   A1*k1cos(k1*L/2) = A2*K2cos(k2*L/2) .. (2)

d)   also, tension in both the strings will be the same
   and wave speed is given by
   v = sqroot(T/mu)
   also, v = w/k

   hence for both the strins
   v1^2*mu1 = v2^2 *mu2
   w^2*mu1/k1^2 = w^2*mu2/k2^2
   mu1/k1^2 = mu2/k2^2
   k2 = k1*sqroot(mu2/mu1)

   A1*sqroot(mu2/mu1)*cos(k1*L/2) = A2cos(k2*L/2)
   A1*sqroot(mu2/mu1)*cos(k1*L/2) = A1*sin(k1*L/2)cos(k2*L/2)/sin(k2*L/2)
   sqroot(mu2/mu1) = tan(k1*L/2)/tan(k2*L/2)
e) mu1 = 4mu*2
   1/2 = tan(k2*sqroot(mu1/mu2)*L/2)/tan(k2*L/2)
   1/2 = tan(k2*L)/tan(k2*L/2)
  
   

Related Questions

Navigate

• Browse (All)
• Browse #
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.