Question

derive an expression for equation of stationary wave on a stretched string​

Answer 1

Answer:

The equation of two waves having the same amplitude, wavelength, and speed but propagating in opposite directions is  

y  

1

​  

=asin  

λ

2π

​  

(vt−x) and

y  

2

​  

=asin  

λ

2π

​  

(vt+x)

Where a is the amplitude, λ is the wave-length and v is the velocity of the wave. A stationary wave is formed due to the superposition of these two waves. The resultant displacement of a particle is given by,

y=y  

1

​  

+y  

2

​  

 

=asin  

λ

2π

​  

(vt−x)+asin  

λ

2π

​  

(vt+x)

Using the relation,

sinC+sinD=2sin  

2

C+D

​  

cos  

2

C−D

​  

,

we have y=2acos  

λ

2π

​  

x⋅sin  

λ

2π

​  

vt

=Asin  

λ

2π

​  

vt

where A=2acos  

λ

2π

​  

x represents the amplitude of the resultant wave.