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Resultant of n Harmonic Waves

Consider n simple harmonic vibrations having equal amplitudes, periods and phases increasing in arithmetic progression by an amount 8. To find the resultant amplitude of such vibrations, a R polygon as shown in Figure 5.5 is constructed. The closing side OP will give the resultant amplitude R and a resultant phase <j>. Resolving the amplitudes parallel and perpendicular to first vibration. We get

 

 

R cos θ = a + a cos δ + … +a cos (n-1) δ        … (1)

R sin θ = 0 +a sin δ + … + a sin (n-1) δ          … (2)

Multiplying equation (1) by (2) sin δ/2 and equation (2) by 2 cos δ/2 one can have respectively:

R cos  θ=a sin n δ/2/sin δ/2  cos  (n-1) δ /2

 

R cos θ =a sin n δ/2/sin δ/2 sin   (n-1) δ /2

And

Squaring and adding, we get

R =A sin n δ/2 /sin δ/2

and on dividing

tan θ tan (n-1) δ /2  or θ =(n-1) δ/2

 

When a and 8 are very small and n is infinitely large such that na remains finite then

taking nδ = 2a and na = t, we get

R =t sin α/α and  θ =nδ /2 =α

 

which are the expression for resultant amplitude and resultant phase.