Unformatted text preview: ∂ 2 p ∂t 2 = c 2 ∇ 2 p , ﬁnd and expression for p in terms of ϕ . Solution: Using the wave equation, we have c 2 ∇ 2 p =B ∇ 2 ∂ϕ ∂t . From this, we then guess that p =B c 2 ∂ϕ ∂t . 3. From the expression you found for p , take partial derivative with respect to time and compare the resulting expression with your previous expression for ∂p ∂t . Does ϕ satisfy the wave equation? If so, what is the wave velocity for ϕ ? Solution: Now take the partial derivative with respect to time of p =B c 2 ∂ϕ ∂t , we ﬁnd ∂p ∂t =B c 2 ∂ 2 ϕ ∂t 2 . Comparing this with the expression we had earlier for ∂p ∂t =B ∇ 2 ϕ , we conclude that ∂ 2 ϕ ∂t 2 = c 2 ∇ 2 ϕ . So ϕ satisﬁes the wave equation with the same velocity c . 1...
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 Fall '04
 POLLACK
 Physics, Magnetism, wave equation, Gradient, ϕ. ∂t ∂t

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