We have discovered a method for selectively detecting high-energy phonons which has enabled us to make detailed measurements of the shapes of high-energy phonon pulses in liquid He-4 at approximate to50 mK. These phonons are created from pulses of low-energy phonons, injected into the helium by a heater. We suppress the detection of the low-energy phonons by orienting a thin film superconducting Zn bolometer at an angle to the phonon beam. We find that the high-energy phonons are created continuously along the propagation path. As there is also velocity dispersion, the signal at any time is due to a range of phonon energies. Computer simulations of the signals from short pulses reproduce the main features of the measured results. The behavior of the high-energy phonon signals as functions of heater power and pulse length are analyzed and this leads us to suggest how the low-energy phonon pulse develops. In the following paper [I. N. Adamenko , Phys. Rev. B 69, 144525 (2004)] the pulse shape for short pulses is derived analytically.