Two superimposing waves are represented by equations

y1= 2sin 2π(10t - 0.4x)

 

y2= 4sin 2π(20t - 0.8x)

 

Find the ratio of Imax and Imin. Ans is (25:9)

 

I know that intensity is proportional to the square of amplitude and

 

maximum amplitude = a1 + a2

 

minimum amplitude = a1 - a2

 

but these equations are only applicable for waves with same frequency.

I also tried plotting it , but I am unable to figure out the value of y1 when y2=4

 

so how do I proceed ? Thanks in advance!

Edited December 9, 20169 yr by snowy

It's been a while but have you plotted y1+y2?

 

Are you familiar with the term beat frequency?

Thanks Klaynos, for your response!

I tried a rough sketch , because from the equations , it can be inferred that T1=2T2

And the resultant amplitude will be max when y2 = 4 , but I'm not sure about y1.

 

Yes! I am aware of beat frequency, it is when 2 waves of nearly equal frequency are superimposed, and the value of it is the difference in frequencies!

It's not asking you about the max of y1 or y2 but the combined maximum. This may be at a different place to the single wave max.

Ok I've just looked at the equations. It is harder than I anticipated as you have unknowns in the form of t and x.

 

So, let's ask you a different question.

 

How do you find the turning points of equations?

By differentiating

dy/dx=0

d2y/dx2 >0 for min

<0 for max

 

PS. I have a restriction of only 2 more posts , so ill keep editing this post to reply.

Edited December 9, 20169 yr by snowy

You need to separate the t and x in the original equations as they are both independent variables.

 

Have you heard of the trigonometric transformation to do this - otherwise known as the sum and difference formulae?

 

https://www.google.co.uk/search?hl=en-GB&source=hp&biw=&bih=&q=sum+and+difference+formulas+in+trigonometry&gbv=2&oq=sum+and+difference+formula+i&gs_l=heirloom-hp.1.3.0i22i30l10.3438.10797.0.13297.28.14.0.14.14.0.187.1622.5j9.14.0....0...1ac.1.34.heirloom-hp..0.28.2310.ngHuv_uG_7k

Thanks Genius for your response!

I tried your advice but one can separate the t and x term only if they had the same amplitude, but here even the amplitude is different!

So where is your working?

 

y1 = 2sin2pi(10t-0.4x)

 

=2sin(20pit - 0.8pix)

 

=2sin(20pit)cos(0.8pix) - 2 cos (20pit)sin(0.8pix)

 

Y2 = 4sin(80pit)cos(1.6pix) - 4cos(80pit)sin(1.6pix)

 

What is the relationship between sin (a) and sin(2pia) ? etc for other multiples of pi and the cos functions?

 

What happens if you form the sum y1 + y2 and collect terms, allowing for the above?