Yes, two waves can have the same amplitude and different frequencies. Mathematically, this can be proven using the superposition principle.
According to the superposition principle, when two or more waves pass through the same point in space, the resulting displacement at that point is the algebraic sum of the individual wave displacements. This principle applies to waves of different frequencies as well.
Let's consider two waves with different frequencies, denoted as f1 and f2, and the same amplitude, denoted as A. Mathematically, we can represent these waves as:
Wave 1: y1 = A * sin(2πf1t) Wave 2: y2 = A * sin(2πf2t)
Where y1 and y2 represent the displacements of the waves at time t.
To find the combined waveform resulting from the superposition of these two waves, we simply add their displacements together:
y = y1 + y2 = A * sin(2πf1t) + A * sin(2πf2t)
Using trigonometric identities, we can rewrite this equation as:
y = 2A * sin((2π(f1+f2)/2)t) * cos((2π(f1-f2)/2)t)
From this equation, we can observe that the combined waveform contains both the sum and difference of the frequencies (f1 + f2) and (f1 - f2). This means that even though the amplitudes of the two waves are the same, their frequencies will influence the resulting waveform. The resulting wave will exhibit variations in its oscillations based on the frequency differences between the two original waves.
Therefore, mathematically, we can prove that two waves with the same amplitude and different frequencies can result in a combined waveform that is influenced by both frequencies.