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An azimuthally symmetric wave (0,t) on a sphere is governed by the wave equation 8²u at² c²a R² sin 800 where k is a constant. sin 8 მო. where t is the time, is the spherical polar angle, c = constant is the speed of the wave and R = constant is the radius of the sphere. (a) Using the separable solution u(e,t) = F(8)G(t), show that F(0) and G(t) obey the following pair of ordinary differential equations: 1 d (sin 6df)+kF = 0, sin e de d²G c²k dt² + R2G=0,

An azimuthally symmetric wave (0,t) on a sphere is governed by the wave equation 8²u at² c²a R² sin 800 where k is a constant. sin 8 მო. where t is the time, is the spherical polar angle, c = constant is the speed of the wave and R = constant is the radius of the sphere. (a) Using the separable solution u(e,t) = F(8)G(t), show that F(0) and G(t) obey the following pair of ordinary differential equations: 1 d (sin 6df)+kF = 0, sin e de d²G c²k dt² + R2G=0,

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4. An azimuthally symmetric wave u(0,t) on a sphere is governed by the wave equation (b) where t is the time, is the spherical polar angle, c = constant is the speed of the wave and R = constant is the radius of the sphere. (c) a²u at² (a) Using the separable solution u(0,t) = F(0)G(t), show that F(0) and G(t) obey the following pair of ordinary differential equations: (d) c² Ә R² sin 0 00 where k is a constant. sin 0 1 d sin e de dF 6 db) + sin 8 +kF = 0, d²G c²k dt² + R² G=0, By a suitable change of variable, show that the equation for F has the form of a Legendre's equation. Then, by considering the boundary condition at the poles (i.e. at 0 = 0 and 0 = π), state the restriction on k. Derive the eigenfunctions, and hence, the general solution of the problem. You may find it useful to let w² = c²k/R², where w≥ 0, for the equation of G (t). Show that u(0,t) = (3 cos² 0 - 1) cos wt is a wave on a sphere satisfying both differential equations in Question 4(a) above. Identify the corresponding values of k and for this solution.
Transcribed Image Text:4. An azimuthally symmetric wave u(0,t) on a sphere is governed by the wave equation (b) where t is the time, is the spherical polar angle, c = constant is the speed of the wave and R = constant is the radius of the sphere. (c) a²u at² (a) Using the separable solution u(0,t) = F(0)G(t), show that F(0) and G(t) obey the following pair of ordinary differential equations: (d) c² Ә R² sin 0 00 where k is a constant. sin 0 1 d sin e de dF 6 db) + sin 8 +kF = 0, d²G c²k dt² + R² G=0, By a suitable change of variable, show that the equation for F has the form of a Legendre's equation. Then, by considering the boundary condition at the poles (i.e. at 0 = 0 and 0 = π), state the restriction on k. Derive the eigenfunctions, and hence, the general solution of the problem. You may find it useful to let w² = c²k/R², where w≥ 0, for the equation of G (t). Show that u(0,t) = (3 cos² 0 - 1) cos wt is a wave on a sphere satisfying both differential equations in Question 4(a) above. Identify the corresponding values of k and for this solution.
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