1. (a) Suppose that the temperature is given by
   T ( x,t) = β x
   where β = 8 deg / meter . At what speed and in what direction (toward positive or
   negative x) would an observer need to move in order to observe a temperature change of
   20 deg / sec?
   (b) Next suppose that the temperature is given by
   T ( x,t) = β x + γ t
   where β is as given above and γ = 2 deg / sec . At what speed and in what direction
   would an observer need to move in order to observe no temperature change at all?
2. In chapter 1 we showed that (assuming the shallow-water limit), the wavenumber of a
   shoaling wave can be predicted from
   ω0 = gH ( x) k ( x).
   Show that the same result is obtained by solving the equation (cf. 6.21)
   cg
   d
   dx
   k ( x) = − ∂Ω
   ∂H
   d
   dx
   H ( x)
   with Ω(k, H ) = gH k .
3. Waves with wavelength 300m approach the coast at an angle of 60 degrees in deep
   water, as shown. What is the corresponding angle in water 5m deep? The full depth
   profile H(x) is unknown, but it does not depend on the longshore coordinate y.
4. Measurements of the bathymetry near a straight beach show that

H(x) = −α x
where

α, the bottom slope, is a positive constant. A wave with frequency

ω0 and
longshore wavenumber

l0 approaches the shore at an angle, as shown. (In the figure,

x1 ≡ x and

x2 ≡ y .)
Show that, in sufficiently shallow water, the rays obey the equation
− dy
d x = l0
gα x
ω0
2
Using the fact that wave crests (or troughs) are everywhere perpendicular to the rays,
write the corresponding equation for wave crests. Solve the equation for the wave crests,
and show that in very shallow water the wave crests have the shape of parabolas, as seen
in the figure. (All of this assumes that waves reach the shoreline without breaking.)
Hint: If two curves are locally perpendicular, then the slope of one equals the negative
reciprocal of the other.

5. An ocean on x<0 has the depth field

H(x, y) = −α x(1+ εcos(βy))
Write down the four coupled equations you would need to solve to determine the path of
a ray, and the values of k and l along it, assuming shallow-water dy