Which wave carries more energy? Which wave carries more energy?
This is a school question (in a course a friend is taking) but I am interested in the answer:

 A: At the same amplitude, higher frequency waves make the rope move faster (because every point has to move the same distance in less time). That means there is more kinetic energy in the wave at higher frequencies. You notice this because it is more tiring to move the rope quickly (that also has to do with the fact that loss mechanisms are stronger at higher frequencies... but even without losses the high frequency wave carries more energy per unit length.
A: There is no definite answer based on longitudinal vs transverse in perfect generality. But consider several examples: 
Seismic P-waves travel more quickly and are longitudinal.
The speed of sound in metal is much, much higher than the speed at which a longitudinal wave travels down, say, a violin string or a piece of rebar.
The speed of sound in water is hundreds of times faster than the speed of ocean waves.
What does speed of wave have to do with energy? The speed indicates the time-scale of response to stimulus. In a Simple Harmonic Oscillator, E = w^2 m x^2. w is the angular frequency - the time scale of the response. Waves are made of coupled harmonic oscillators, and in mechanical systems, the harmonic oscillation is formed BY the coupling, so the frequency of the oscillation also gives the strength of the coupling.
SO, for a fixed amplitude, longitudinal waves generally carry more energy. Not a big surprise - I can bend a 5 meter bar of rebar 1 cm to the side (transverse displacement) without effort. I can't compress it (longitudinal displacement) by even 1 mm even if I get out hand tools.
There are definitely ways around this - sometimes the coupling between neighbors isn't what causes the oscillation in the first place. Sometimes something else happens to change it. This is not perfectly general, to repeat what I said up front. But there is this tendency in mechanical waves.
That said, I don't think your homework question was asking that.
