How to interpret this Wave Equation (Derivation) Help ?

[Shahab Mirza]

Hi dear people , Hello
I waw studying super position of two Sound Waves , traveling in same medium with same frequency , same wavelength and same amplitude while differing in phase .
quick derivation :
Wave 1 displacement y1= A sin (kx-vt ) and wave 2 displacement y2= A sin (kx-vt-phase constant)
after adding y1+y2 and then using trigonometry I got Final equation :-

y= (2A cos phase constant / 2) sin (kx-vt - phase constant / 2 ) ok now my question is that how it is interpreted in my textbook ? it is written that this equation shows that . 1. It can bee seen that resultant wave function is hormonic and has same frequency and wavelength as individual waves . Amplitude is twice than individual wave of same wavelength .

I want to know that How people interpret these equations and develop relationship between them ? I am having difficulty in physics because I lack this ability to know what equation delivers , Thanks please help me out

 

[ash64449]

Amplitude is twice than individual wave of same wavelength

Not always. It depends on the phase constant.

The value of cos(angle) should be 1 if the amplitude is to be 2A. If phase constant is 0,then cos0=1 and amplitude is 2A.

 

[Shahab Mirza]

Not always. It depends on the phase constant.

The value of cos(angle) should be 1 if the amplitude is to be 2A. If phase constant is 0,then cos0=1 and amplitude is 2A.

Thanks for your reply Sir ,
Actually I used some of my efforts to understand this phenomenon on my intermediate level .
Below is what I understood .
2A (cos phase / 2) in this equation if phase is 0 then cos = 1 then amplitude is maximum means twice than individual original wave. and integral multiple of 0,2pi.3pi and so on and resulting in constructive interference.
And when if phase is pi or any odd multiple of pi ,then cos angle value = 0 and there will be minimum amplitude resulting in destructive interference.

My conclusions are :-
1: If interference is Constructive then amplitude of 1st 2nd and s one overtones/hormonics forms larger dislplacments than original individual wavelength , means if 3 waves are traveling in phase means one after other than will make 3 times larger amplitude.
2: If interference is destructive than waves will be out of phase and crest of one will fall on trough of other and there amplitudes will be equal on both sides hence cancelling each others displacement at each point .and resultant wave will have zero amplitude .

 

[ash64449]

Your conclusions are correct.

 

[Shahab Mirza]

Your conclusions are correct.

Thats good , Thanks a lot Sir