At time t=0 an elastic string is stretched to a line density
ho and a tension T between the lines
at x=0 and x=L in the (x,y)-plane, where
ho ,T and L are positive constants. The small
transverse displacement y(x,t) satisfies the wave equation
(del^(2)y)/(delt^(2))=c^(2)(del^(2)y)/(delx^(2)), for ,00
where the wave speed c=sqrt((T)/(
ho )). The end of the string at x=L is fixed so that y(L,t)=0 for
t>0. The other end is attached to a small ring of mass M which can move freely on a smooth
wire lying along the y-axis.
(a) Assuming that the effects of gravity and air resistance are negligible, write down the y -
component of Newton's Second Law for the ring and deduce that, to a first approximation
for |y_(x)|≪1,
M(del^(2)y)/(delt^(2))(0,t)=Ty_(x)(0,t) for t>0.
(b) Let omega be a positive constant and epsi a constant. Show that there is a non-trivial separable
solution of the form y(x,t)=F(x)sin(omega ct+epsi ) only if omega is a root of the equation
tan(omega L)=(alpha )/(omega L)
where alpha is a dimensionless parameter that you should determine.
(c) The energy of the system is given by
E(t)=(
ho )/(2)int_0^L ((dely)/(delt))^(2)dx+(T)/(2)int_0^L ((dely)/(delx))^(2)dx+(M)/(2)((dely)/(delt)(0,t))^(2).
(i) State the physical significance of each of the three terms on the right-hand side of this
equality and show that E is constant.
(ii) Deduce that there is at most one solution of the initial boundary value problem for
y(x,t) given by the wave equation and boundary conditions above, subject to the given
initial conditions y(x,0)=f(x) and y_(t)(x,0)=g(x) for 0.
1. At time t = 0 an elastic string is stretched to a line density p and a tension T between the lines at x = 0 and x = L in the (x,y)-plane, where , T and L are positive constants. The small transverse displacement y(c,t) satisfies the wave equation
O2y Ot2
202y dx2
0<7T>x>0JOJ
where the wave speed c= VT/p. The end of the string at x = L is fixed so that y(L,t) = 0 for t > 0. The other end is attached to a small ring of mass M which can move freely on a smooth wire lying along the y-axis.
(a) Assuming that the effects of gravity and air resistance are negligible, write down the y- component of Newton's Second Law for the ring and deduce that, to a first approximation for|yx|< 1,
(0,t)=Tyx(0,t) fort>0 a+2
b) Let w be a positive constant and & a constant. Show that there is a non-trivial separable solution of the form y(x,t) = F(x) sin(wct + e) only if w is a root of the equation
a tan(wL)= WL'
where a is a dimensionless parameter that you should determine.
(c) The energy of the system is given by
T + xp 2
2 (Oy M dx+ dx 2
E(t
10
(i) State the physical significance of each of the three terms on the right-hand side of this equality and show that E is constant. (ii) Deduce that there is at most one solution of the initial boundary value problem for y(,t) given by the wave equation and boundary conditions above, subject to the given initial conditions y(,0) = f(x) and yt(x,0) = g(x) for 0< x < L.