The equation relating tension and speed of wave is given by:
#v=sqrt(T/mu)#
where: #T" is the tension"; mu" is the linear density of the string"#
Since we know that #v=flambda#, we can rewrite the equation into:
#flambda=sqrt(T/mu)#
I)Tension increased by a factor of 4
Let #T# be the initial tension of the string
#:.4T# will be the final tension of the string
#4#loops are formed,
#L=2lambda#
The string used is the same, thus the linear density, #mu# is a constant.
The frequency of the vibrator is also kept as the same, #f# is another constant.
Rearranging
#flambda=sqrt(T/mu)#
#f^2mu=T/lambda^2#
since #f# and #mu# are constants, their product is a constant too.
#T_i/lambda_i^2=T_f/lambda_f^2#
subscript #i# for initial, #f# for final
#T/(L/2)^2=(4T)/lambda_f^2#
#(4T)/(4(L^2/4))=(4T)/lambda_f^2#
#4(L^2/4)=lambda_f^2#
#lambda_f^2=L^2#
#lambda_f=L#
Therefore the number of loops formed will be 2.
II)Frequency increased by a factor of 4
Now, Tension instead of frequency becomes constant.
#flambda=sqrt(T/mu)#
Since Tension and Linear density remain the same, #sqrt(T/mu)# is constant.
Thus,
#f_ilambda_i=f_flambda_f#
Let #f# be the initial frequency,
#4f# will be the final frequency.
#4#loops are formed,
#L=2lambda#
#lambda=L/2#
#f(L/2)=(4f)lambda_f#
#lambda_f=f(L/2)/(4f)#
#lambda_f=L/8#
Thefore, the number of loops is 16.
