The following equations represent progressive transverse waves $Z_1=A\cos (\omega t-kx)$, $Z_2=A\cos (\omega t+kx)$, $Z_3=A\cos (\omega t+ky)$ and $Z_4=A\cos (2\omega t-2ky)$. A stationary wave will be formed by superposing :

(1) Z₁ and Z₂

(2) Z₁ and Z₄

(3) Z₂ and Z₃

(4) Z₃ and Z₄

Two traveling waves $y_1=A\sin \left[k\right(x-ct\left)\right]$ and $y_2=A\sin \left[k\right(x+ct\left)\right]$ are superimposed on the string. The distance between adjacent nodes is :

(1) ct / π

(2) ct / 2π

(3) π / 2k

(4) π / k

A string fixed at both ends is vibrating in two segments. The wavelength of the corresponding wave is :

(1) $\frac{l}{4}$

(2) $\frac{l}{2}$

(3) l

(4) 2l

A 1 cm long string vibrates with the fundamental frequency of 256 Hz. If the length is reduced to $\frac{1}{4}cm$  keeping the tension unaltered, the new fundamental frequency will be :

(1) 64

(2) 256

(3) 512

(4) 1024

Standing waves are produced in a 10 m long stretched string. If the string vibrates in 5 segments and the wave velocity is 20 m/s, the frequency is :

(1) 2 Hz

(2) 4 Hz

(3) 5 Hz

(4) 10 Hz

A string is producing transverse vibration whose equation is $y=0.021\sin (x+30t)$, Where x and y are in meters and t is in seconds. If the linear density of the string is 1.3×10^–4 kg/m, then the tension in the string in N will be :

(1) 10

(2) 0.5

(3) 1

(4) 0.117

A stretched string of length l, fixed at both ends can sustain stationary waves of wavelength λ, given by 

(1) $\lambda =\frac{n^2}{2l}$

(2) $\lambda =\frac{l^2}{2n}$

(3) $\lambda =\frac{2l}{n}$

(4) $\lambda =2ln$

A string on a musical instrument is 50 cm long and its fundamental frequency is 270 Hz. If the desired frequency of 1000 Hz is to be produced, the required length of the string is :

(1) 13.5 cm

(2) 2.7 cm

(3) 5.4 cm

(4) 10.3 cm

The tension in a piano wire is 10N. What should be the tension in the wire to produce a note of double the frequency :

(1) 5 N

(2) 20 N

(3) 40 N

(4) 80 N

A string of 7 m length has a mass of 0.035 kg. If the tension in the string is 60.5 N, then the speed of a wave on the string is :

(1) 77 m/s

(2) 102 m/s

(3) 110 m/s

(4) 165 m/s