# Superposition of harmonic oscillations

• Lengalicious
In summary, the task is to find the amplitude and phase shift of two superposed harmonic oscillations, given by x1(t)=3sin(2∏t+∏/4) and x2(t)=3cos(2∏t). The first function can be converted to cosine form, making its phase angle -π/4. The two functions can be represented as phasors and added, resulting in a magnitude calculated using the cosine law or the component method.
Lengalicious

## Homework Statement

Find the amplitude and phase shift of the following two superposed harmonic oscillations.

## Homework Equations

x1(t)=3sin(2∏t+∏/4)
x2(t)=3cos(2∏t)

## The Attempt at a Solution

Ok normally i would be able to do this, however one oscillation is cos and the other sin, so i can't use the trig identity for sin+sin or cos+cos. Is it possible to turn 3sin(2∏t+∏/4) into 3cos(2∏t+∏/4+∏/2) or is that invalid?

Lengalicious said:

## Homework Statement

Find the amplitude and phase shift of the following two superposed harmonic oscillations.

## Homework Equations

x1(t)=3sin(2∏t+∏/4)
x2(t)=3cos(2∏t)

## The Attempt at a Solution

Ok normally i would be able to do this, however one oscillation is cos and the other sin, so i can't use the trig identity for sin+sin or cos+cos. Is it possible to turn 3sin(2∏t+∏/4) into 3cos(2∏t+∏/4+∏/2) or is that invalid?

I think you mean -π/2 there, but otherwise yep, that's fine. (Or there are similar trig identities you could use, but of course it will add up to the same thing.)

Yeh -∏/2 is what i meant, one more question when calculating the amplitude i find that somehow the following equation is derived, A=√(A12+A22+2A1A2cos(θ21)). I don't understand how this is found?

Bump: How do I find the amplitude, i get a phase of pi/8 and an amplitude of 5.5 but i think my amplitude is wrong, can someone check please?

Lengalicious said:
Yeh -∏/2 is what i meant, one more question when calculating the amplitude i find that somehow the following equation is derived, A=√(A12+A22+2A1A2cos(θ21)). I don't understand how this is found?

Lengalicious said:
Bump: How do I find the amplitude, i get a phase of pi/8 and an amplitude of 5.5 but i think my amplitude is wrong, can someone check please?

The formula above is an application of the cosine law. You can see how it applies if you consider that since both have the same frequency ##(2\pi)##, the two functions can be represented in phasor form and added as phasors.

Convert the first sin function to cosine as you have suggested previously. That makes its phase angle ##-\pi/4##. Its phasor is then a vector of length 3 with angle ##-\pi/4## from the x-axis. The second function has no phase, so it's a vector of length 3 on the x-axis. Add as you would any two vectors. You'll find that one way to find the magnitude is to apply the cosine law -- draw it, you'll see. You could also add them via the component method. Either works just fine.

Ah, that helps a lot thanks very much

## What is superposition of harmonic oscillations?

Superposition of harmonic oscillations is the principle that states when two or more harmonic oscillations occur simultaneously, the resulting motion is the sum of the individual motions. In simpler terms, it is when two or more waves overlap and combine to create a new wave.

## What are harmonic oscillations?

Harmonic oscillations are repetitive motions or vibrations that occur at a constant frequency and amplitude. They are characterized by their sinusoidal shape and can be found in various systems such as pendulums, springs, and electrical circuits.

## What is the equation for superposition of harmonic oscillations?

The equation for superposition of harmonic oscillations is y = A1sin(w1t) + A2sin(w2t), where A1 and A2 represent the amplitudes of the individual oscillations and w1 and w2 represent their corresponding angular frequencies.

## How does superposition of harmonic oscillations apply to real-world situations?

Superposition of harmonic oscillations can be seen in many real-world situations, such as sound waves combining to create music, ocean waves overlapping to form larger waves, and the interference patterns in light waves. It is a fundamental principle in understanding and analyzing wave phenomena.

## What is the difference between constructive and destructive interference in superposition of harmonic oscillations?

Constructive interference occurs when two waves with the same frequency and amplitude overlap and add together to create a larger amplitude. In contrast, destructive interference occurs when two waves with opposite amplitudes overlap and cancel each other out, resulting in a smaller amplitude or no wave at all.

• Introductory Physics Homework Help
Replies
9
Views
278
• Introductory Physics Homework Help
Replies
16
Views
585
• Introductory Physics Homework Help
Replies
3
Views
381
• Introductory Physics Homework Help
Replies
10
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
228
• Introductory Physics Homework Help
Replies
4
Views
391
• Introductory Physics Homework Help
Replies
12
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
172
• Introductory Physics Homework Help
Replies
6
Views
921
• Introductory Physics Homework Help
Replies
8
Views
821