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Unformatted text preview: 32A - HOMEWORK 7 SOLUTIONS ♣ PROBLEM 3.1. The proof for ϕ is in PARTIAL DIFFERENTIAL EQUATIONS. The proof for ψ is ∂ψ ( x − ct ) ∂t = ψ ( x − ct ) ∂ψ ( x − ct ) ∂t = − cψ ( x − ct ) . Differentiating once again we obtain ∂ 2 ψ ( x − ct ) ∂t 2 = − cψ ( x − ct ) ∂ψ ( x − ct ) ∂t = c 2 ψ ( x − ct ) . (1) On the other hand, ∂ψ ( x − ct ) ∂x = ψ ( x − ct ) ∂ψ ( x − ct ) ∂x = − ψ ( x − ct ) . thus, differentiating again, ∂ 2 ψ ( x − ct ) ∂x 2 = − ψ ( x − ct ) ∂ψ ( x − ct ) ∂x = ψ ( x + ct ) . (2) Combining (1) and (2) the result for ψ ( x − ct ) follows. ♣ PROBLEM 3.3. Equality (3.8) says that v ( t, x ) = ϕ ( x + ct ) − ϕ ( x − ct ) (3) thus the fact that v ( t, x ) satisfies the wave equation is a consequence of Problem 3.1 (with ϕ = ψ ) . ♣ PROBLEM 3.4. It follows from (3) that ∂v ( t, x ) ∂t = cϕ ( x + ct ) + cϕ ( x − ct ) and we have ϕ ( x ) = w ( x ) , thus ∂v ( t, x ) ∂t = cw ( x + ct ) + cw ( x − ct ) . (4) ♣ PROBLEM 3.5. That the first term in u ( t, x ) = u ( x + ct ) + u ( x − ct ) 2 + 1 2 c Z x + ct x − ct u 1 ( ξ ) dξ (5) satisfies the wave equation was proved in Problem 3.1; for the second term we use Problem 3.3. We only have to check the initial conditions. The first initial condition is u (0 , x ) = u ( x ) + u ( x ) 2 + 1 2 c Z x...
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This note was uploaded on 06/03/2011 for the course MATH 32A 32A taught by Professor Moshchovakis during the Spring '10 term at UCLA.
- Spring '10