Why Is the Resultant Amplitude of Interfering Waves Not Simply A1 + A2?

[desmond iking]

Homework Statement

when a point is intefered by 2 waves of different phase , the resultant is y1 + y2 ... but why the resultant amplitude can't be = A1 + A2 ... but is sqrt root ((A1)^2 + (A2)^2) ??

this is actually a online note.

Homework Equations

The Attempt at a Solution

 

[NascentOxygen]

when a point is intefered by 2 waves of different phase , the resultant is y1 + y2 ... but why the resultant amplitude can't be = A1 + A2 ... but is sqrt root ((A1)^2 + (A2)^2) ??

It isn't, and your pic shows it isn't. The peak of the resultant amplitude is determined using the cosine rule and the phase angle difference. Only if the phase angle was 90 degrees would your equation using Pythagoras hold.

 

[desmond iking]

It isn't, and your pic shows it isn't. The peak of the resultant amplitude is determined using the cosine rule and the phase angle difference. Only if the phase angle was 90 degrees would your equation using Pythagoras hold.

sorry, i mean why can't i add up A1 and A2 to get the resultant amplitude... tHat means resultant amplitude =A1+ A2

 

[dauto]

Because y₁ and y₂ never reach their maximum values at the same time. unless they're in phase (θ = 0).

 

[HalfLostAndSearching]

The phenomenon of superposition of waves is a fundamental concept in the field of physics that describes the behavior of waves when they interact with each other. When two waves of different phase interfere with each other, the resulting amplitude is not simply the sum of the individual amplitudes, but rather the square root of the sum of the squares of the individual amplitudes. This is known as the principle of superposition.

To understand why this is the case, we must first understand the nature of waves. Waves are disturbances that travel through a medium, such as air or water. They have a certain amplitude, which is the maximum displacement of the medium from its rest position, and a certain wavelength, which is the distance between two consecutive peaks or troughs of the wave.

When two waves of different phase meet, they combine to form a new wave. The amplitude of this new wave is determined by the sum of the individual amplitudes at each point where the waves intersect. However, since the two waves are out of phase, they cancel each other out at some points and reinforce each other at others. This results in a wave with a varying amplitude along its length.

To find the overall amplitude of this new wave, we must use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the hypotenuse represents the overall amplitude of the new wave, while the other two sides represent the individual amplitudes of the two interfering waves.

Therefore, the resulting amplitude is not simply the sum of the individual amplitudes, but rather the square root of the sum of the squares of the individual amplitudes. This is why we use the formula sqrt((A1)^2 + (A2)^2) to calculate the resultant amplitude in cases of wave interference.

In conclusion, the principle of superposition is a fundamental concept in the study of waves that explains the behavior of waves when they interact with each other. The resulting amplitude of two interfering waves is determined by the sum of the squares of the individual amplitudes, which is why we use the square root formula to calculate it.