Yes, it is possible for one wave to cancel another wave so that their combined amplitude is zero. This phenomenon is known as wave interference, specifically destructive interference.
Destructive interference occurs when two waves of equal frequency and opposite phase meet. In other words, if one wave has a positive displacement at a particular point in time, the other wave has an equal negative displacement at that same point in time. When these waves combine, the positive and negative displacements cancel each other out, resulting in a net displacement of zero.
Mathematically, destructive interference can be understood by considering the principle of superposition. When two waves interfere, the amplitudes of the individual waves are added together at each point in space and time. If the two waves have the same amplitude and are perfectly out of phase, their sum will be zero at all points where they overlap.
Destructive interference is commonly observed in various situations. For example, if you have two speakers emitting sound waves with the same frequency but with opposite phases, you may notice areas in the room where the sound seems to be significantly reduced or canceled out. This occurs because the waves from the two speakers interfere destructively at those particular locations.
It's important to note that destructive interference occurs when specific conditions are met, such as equal frequencies, opposite phases, and proper alignment. In most cases, waves do not perfectly cancel each other out, but the resulting amplitude may be significantly reduced due to partial cancellation.