Please help ! Superposition of waves

[snowy]

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, 2016 by snowy

[Klaynos]

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

 

Are you familiar with the term beat frequency?

[snowy]

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!

[Klaynos]

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?

[snowy]

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, 2016 by snowy

[studiot]

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

[snowy]

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!

[studiot]

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?