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SIO 111/Physics 111
Winter 2016
Homework #3
Due Wednesday, January 27, 2016
1. Sketch the function
⎧ 1
,
⎪
f ( x ) = ⎨ 2d
⎪ 0,
⎩
x <d
x >d
where d is a constant. Then use (3.14) to compute its Fourier transform, a(k) and b(k).
Sketch the functions a(k) and b(k). In what sense is the width of f(x) inversely
proportional to the width of a(k)?
2. A general theorem (called Parseval’s theorem) related to Fourier transforms states
that
∞
∫ f ( x)
2
dx = π
−∞
∞
∫ (a(k )
0
2
)
+ b( k ) dk
2
Verify this theorem for the function f ( x ) = e− β x , whose Fourier transform is given by
(3.17) and (3.18).
2
3. As we have shown, the wave field corresponding to the initial conditions
η( x,0) = F ( x )
and
∂η
( x,0) = G( x )
∂t
is given by
η( x,t ) =
+∞
∫ dk [ A(k ) cos(kx − ω (k )t ) + B(k ) sin(kx − ω (k )t )]
−∞
where
1
A( k ) =
2π
+∞
1 +∞
∫ dx F ( x ) cos(kx ) + 2πω (k ) ∫ dx G( x ) sin(kx )
−∞
−∞
and
1
B( k ) =
2π
+∞
1 +∞
∫ dx F ( x ) sin(kx ) − 2πω (k ) ∫ dx G( x ) cos(kx ) .
−∞
−∞
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What properties of the wave amplitudes A(k) and B(k) result when the initial conditions
have the properties given below?
Example: F ( x ) = F (−x )
and
G( x ) = 0
G( x ) = 0
Answer: A( k ) = A(−k )
(a)
F ( x ) = −F (−x )
and
(b)
F ( x) = 0
and
G( x ) = G(−x )
Answer:
(c)
F ( x) = 0
and
G( x ) = −G(−x )
Answer:
and
B( k ) = 0
Answer:
4. In each case of Problem 3, suppose that the surface elevation at x = L , namely η( L,t )
is measured to be f(t). What then is the surface elevation at x = −L ?
Example: If A( k ) = A(−k ) and B( k ) = 0 , then
η( L,t ) =
+∞
∫ dk A(k ) cos(kL − ω (k )t ) ≡ f (t)
−∞
On the other hand,
η(−L,t ) =
=
+∞
∫ dk A(k ) cos(−kL − ω (k )t )
−∞
+∞
∫ dk A(−k ) cos(kL − ω (−k )t )
−∞
=
+∞
∫ dk A(k ) cos(kL − ω (k )t )
−∞
= f (t)
where the second step corresponds to a change in the (dummy) integration variable.
5. We have shown that the general solution to the initial problem is
η( x,t ) =
+∞
∫ dk [ A(k ) cos(kx − ω (k )t ) + B(k ) sin(kx − ω (k )t )]
−∞
where the function ω ( k ) corresponds to the dispersion relationship. Suppose that the
dispersion relationship takes the particular form
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ω ( k ) = gH k ≡ c 0 k
That is, suppose that the shallow-water limit can be used for all wavenumbers. Show
that, in this case, the general solution reduces to the form
η( x,t ) = R( x − c 0 t ) + S ( x + c 0 t )
and state how the functions R and S are related to the functions A(k) and B(k).
Hint: Split the integral over all k into an integral over positive k and an integral over
negative k.
6. Assuming that the wave field does indeed take the form
η( x,t ) = R( x − c 0 t ) + S ( x + c 0 t )
determine the functions R and S that correspond to the initial conditions
η( x,0) = 0
and
∂η
( x,0) = Dsin(k0 x )
∂t
where D and k0 are constants.