Stationary Waves: Finding Minimum Intensity Distance

[d0h]

stationary waves, urgent!

helllloo.

i'm having a major exam tomorrow and i stumbled across this question just now..would appreciate all your help!

Two loudspeakers S1 and S2 are placed a distance 4.0m apart facing each other. The loudspeakers produce sound waves of frequency 165 Hz in phase with one another. A microphone, connected to a CRO, is moved along a straight line joining S1 and S2. The sound received by the microphone fluctuates regularly. Speed of sound is 330m/s.

(ii) Find the shortest distance from S1 where a minimum intensity is detected.

I understand that a stationary wave is formed between the 2 speakers and what this question requires is the distance of the first node from S1. Not too sure on how to get there though..

the answer key provided just drew 2 cosine waves from S1 to S2, one negative and the other positive. since the wavelength is 2.0m, the min. intensity is detected at 0.5m..

what i don't understand is how and why the cosine waves are derived, why not sine waves?

thanks!

 

[Astronuc]

A cosine wave is a sine wave with a different phase angle, i.e. cosine wave is a sine wave shifted by pi/2, i.e. cos x = sin (x+pi/2).

Use the same wave form (sin or cos) for both speakers, i.e. the speakers are in phase.

The minimum occurs where the peak (max) of one wave cancels the trough (min) of the other.

 

[d0h]

hmm..

if there are 2 sine curves, the first crest and trough would be 0.5m from S1..
but for cosine curves, the point S1 itself would start with a crest and a trough, the next being 1.0m away

how do i determine if it starts with a sine or a cosine?

 

[Astronuc]

The issue is not whether the source is a sine or cosine wave, but rather the phase difference between the two sources, both temporally (timewise) and spatially.

The example provided,

the answer key provided just drew 2 cosine waves from S1 to S2, one negative and the other positive.

indicates that the sources are temporally out of phase by half a wavelength, but they are spatially separated by 2 wavelengths, the minimum intensity is at either source.

If one uses the same waves for both sources, which are also separated by an integral number (integer) of wavelengths, then they cancel at a distance of one half-wavelength from the either source.