The energy carried by a linear wave is proportional to the square of its amplitude. In this scenario, we have two waves with the same amplitude but different wavelengths. Let's assume the longer wavelength wave has a wavelength of λ and the shorter wavelength wave has a wavelength of λ/2.
The energy carried by a wave is also proportional to the square of its frequency. The frequency of a wave is inversely proportional to its wavelength, so the longer wavelength wave has a lower frequency (f) compared to the shorter wavelength wave.
Now, let's consider the wave equation:
v = f * λ
where: v is the speed of the wave, f is the frequency of the wave, and λ is the wavelength of the wave.
Since the waves have the same speed (v) and the shorter wavelength wave has a wavelength of λ/2, we can write the equation for their frequencies as follows:
v = f₁ * λ v = f₂ * (λ/2)
From these equations, we can see that f₂ = 2 * f₁.
Now, let's compare the energies carried by these waves. The energy (E) of a wave is proportional to the square of its amplitude (A) and the square of its frequency (f):
E = A² * f²
Since the waves have the same amplitude, we can compare their energies by comparing their frequencies squared:
(E₂ / E₁) = (f₂² / f₁²) = (2 * f₁)² / f₁² = 4 * (f₁ / f₁)² = 4
Therefore, the shorter wavelength wave transmits 4 times more energy than the longer wavelength wave.