# Sine Wave Addition: Standing Waves?

• marla11
In summary, two sine waves with the same frequency and amplitude but different phase shifts can still produce a standing wave. The addition of the two waves results in a waveform that fits the standing wave equation, and the phase shift does not affect this.

#### marla11

If two sine waves have the same frequency and amplitude but have different phase shift do they still produce a standing wave?
Thanks for the help.

As far as i can see it doesn't say anything about phase shift so does that mean it doesn't affect anything?

wt is the phase shift

see http://en.wikipedia.org/wiki/Phase_(waves [Broken])

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I'm sorry but I'm still confused. If y1 = Asin(kx-wt+phi) and y2 = Asin(kx+wt)
the addition is y= 2Acos(wt-phi/2)sin(kx+phi/2) where phi is the phase shift between 0 and 2pi. Does this still fit the standing wave equation y=(2Asin(kx))cos(wt) meaning its a standing wave or does the difference in phase shift mean they do not create a standing wave?

You have one forward-traveling wave (wt-kx) and one backward wave (wt+kx) of the same amplitude, which is a standing wave. My CRC Math Tables (10th Ed, 1954) on page 345 shows the sum

sin(x) + sin(y) = 2·sin[(x+y)/2]·cos[(x-y)/2]

Bob S

Marla,

Your equations will be easier to read if you typeset them in LaTeX.

Yes, the equation you give is a standing wave. If you start with

$\Psi(x,t) = A\cos(\omega \left[t-t_0\right]) \sin (k\left[x-x_0\right])$

you can just define a new time coordinate and new space coordinate by

$t' = t - t_0$
$x' = x - x_0$.

Then your original equation is just

$\Psi(x',t') = A \cos(\omega t')\sin(k x')$,

showing that the waveform is exactly the same as the standing wave you're used to.

## 1. What is a sine wave and how is it used in standing wave analysis?

A sine wave is a type of periodic function that describes a smooth and repetitive oscillation. In standing wave analysis, sine waves are used to represent the oscillating displacement of particles in a medium, such as a string or air column.

## 2. How do you add sine waves to create standing waves?

To create a standing wave, two or more sine waves with the same frequency and amplitude are added together. The resulting wave will have points of constructive and destructive interference, creating a stationary pattern.

## 3. What is the relationship between the wavelength and frequency of a standing wave?

The wavelength of a standing wave is equal to twice the length of the medium, divided by the number of nodes (points of zero displacement). The frequency of a standing wave is determined by the speed of the wave and the wavelength, and is inversely proportional to the wavelength.

## 4. How is the amplitude of a standing wave determined?

The amplitude of a standing wave is determined by the amplitudes of the individual sine waves that are added together. The amplitude at each point of the standing wave is the sum of the amplitudes of the component waves at that point.

## 5. What are some real-world applications of standing waves?

Standing waves have many practical applications, such as in musical instruments, where they produce distinct tones and harmonics. They are also used in medical imaging, such as ultrasound, and in telecommunications for signal transmission. Standing waves can also be found in natural phenomena, such as ocean waves and seismic waves.