A wave of amplitude 0.30m interferes with a second wave of amplitude 0.20m traveling in the same direction. What are (a) the largest and (b) the smallest resultant amplitudes that can occur, and under what conditions will these maxima and minima arise?
Well, when two waves collide, it's like a battle of amplitudes! The largest resultant amplitude, or "The Hulk" of amplitudes, occurs when the two waves perfectly align to create maximum constructive interference. It's like a team of synchronized swimmers performing a flawless routine. In this case, the largest resultant amplitude will be the sum of the individual amplitudes, so we have 0.30m + 0.20m = 0.50m. Voila! Just make sure those waves are perfectly in sync, like a boy band that's been rehearsing for years.
Now, for the smallest resultant amplitude, or "The Timid Mouse" of amplitudes, we have the waves working against each other in destructive interference. It's like trying to walk in opposite directions on a really narrow sidewalk while carrying a plate of spaghetti. The smallest resultant amplitude occurs when the two waves perfectly cancel each other out. In this case, the smallest resultant amplitude will be the difference between the individual amplitudes, so 0.30m - 0.20m = 0.10m. Ta-da! The waves say "nope" to each other and decide to call it a day.
Remember, to get these maxima and minima, the waves need to be perfectly in phase for constructive interference and perfectly out of phase for destructive interference. It's like finding the right dance partner, but for waves. Keep on wavin'!
To find the largest and smallest resultant amplitudes in wave interference, we need to consider the principle of superposition.
When two waves interfere constructively, their amplitudes will add up, resulting in a larger amplitude. When they interfere destructively, their amplitudes will subtract, resulting in a smaller amplitude.
(a) The largest resultant amplitude occurs when the two waves interfere constructively. In this case, the amplitudes of the two waves add up.
Resultant amplitude = Amplitude of first wave + Amplitude of second wave
Resultant amplitude = 0.30m + 0.20m
Resultant amplitude = 0.50m
Therefore, the largest resultant amplitude is 0.50m.
(b) The smallest resultant amplitude occurs when the two waves interfere destructively. In this case, the amplitudes of the two waves subtract.
Resultant amplitude = Amplitude of first wave - Amplitude of second wave
Resultant amplitude = 0.30m - 0.20m
Resultant amplitude = 0.10m
Therefore, the smallest resultant amplitude is 0.10m.
The conditions for these maxima and minima to arise depend on the phase difference between the two waves.
- Constructive interference occurs when the waves are in phase, or their phase difference is equal to an integer multiple of 2π.
- Destructive interference occurs when the waves are out of phase, or their phase difference is equal to an odd multiple of π.
So, to achieve the largest resultant amplitude, the two waves need to have a phase difference of 0 radians or any integer multiple of 2π radians.
To achieve the smallest resultant amplitude, the two waves need to have a phase difference of π radians or any odd multiple of π radians.
To find the largest and smallest resultant amplitudes, we need to consider the constructive interference and destructive interference conditions.
Constructive Interference:
Constructive interference occurs when two waves are in phase, meaning their crests and troughs align. In this case, the waves add up and result in a larger amplitude. To calculate the largest resultant amplitude, we simply add the amplitudes of the two waves:
Largest Resultant Amplitude = Amplitude of Wave 1 + Amplitude of Wave 2
= 0.30m + 0.20m
= 0.50m
Thus, the largest resultant amplitude is 0.50m, and it occurs when the waves are in phase and align perfectly.
Destructive Interference:
Destructive interference occurs when two waves are out of phase, meaning their crests and troughs cancel each other out. In this case, we subtract the amplitudes of the two waves to find the smallest resultant amplitude:
Smallest Resultant Amplitude = |Amplitude of Wave 1 - Amplitude of Wave 2|
= |0.30m - 0.20m|
= 0.10m
Thus, the smallest resultant amplitude is 0.10m, and it occurs when the waves are completely out of phase, with the crest of one wave aligning with the trough of the other wave.
In summary, the largest resultant amplitude is 0.50m, occurring when the waves are in phase. The smallest resultant amplitude is 0.10m, occurring when the waves are completely out of phase, canceling each other out.