Question

The wave equation can be written as:∂^2 y/∂x^2 = 1/v^2 (∂^2 y /∂t^2)

where y = y(x,t) has units of meters, x is also in meters, and t is in seconds. (a) Show explicitly that the function y(x,t) = ymsin(kx)cos(wt) satisfies the wave equation (6 points). (b) Is the function for y = y(x,t) describe a traveling wave? You must explain your answer to get full credit (2 points). 8. On a winter day with a temperature of Tc, the temperature inside a room is Th, where Th > Tc. The heat conduction rate through a slab of thickness L, surface area A, thermal conductivity k is measured to be H. Assume you found a way to triple the slab thickness and reduce the thermal conductivity to one quarter, what is the new heat transfer rate H’ in terms of H? Assume the temperatures inside and outside the room and the surface area remains the same (4 points).

Answer 1

dx? V2 Wave Equation : oly - 1 day at² (a) To show explicitly 9Cx, t) = Ym sin(kx) cos(wt) satisfies wave equation Ans - y(x, t) - Ym sin(kx) cos (wt) 24 = kym cos(kx) Cos(wt) 324 =-Kym sin(kx) (05(wt) = -kºy(2, +) Jx dx? at dt2 Now dy --wym sin(kx) sin(wt) dly --w?ym sin(kx) cos(wt) = -w2yCx,t) — since I get a = 1 / 2 [w²y (e,t)] = - k² y (x, t) – 3 As we know = w or V = 4 08, W2 = K? From eqh 1 and ② we can say the function yCae,+ = Ym sin(kx) cos(wt) Satisfies the wave equation (b) yes the function yo y(x, t) describe a wave. As the function depends on both space part and time part. It means the wave is sinusoidal in nature. ty travelling 14 Sinka Coswt 1/2 ST/2 34 34 /27 134 2

Temperature inside a room is The Given: Slab of thickness L , Surface Area A thermal conductivity Ki Assuming L'=3L and K'= k/4. To calculate: New heat transfer rate H in terms of H. We know Heat transfer Rate (H)= KA ( Ti - Tc) So, New Heat transfer Rate (H') = K'A (Tn - Tc) L! 08, H'- (K/4). A (In-Tc) (34) (08 08, H = KA (Tn - Tc) 1 x KA (Th-Tc) Х 12 L 12 L 00, H = H 12