A transverse harmonic wave on a string is described by
$y\left(x,t\right)=3.0sin(36t+0.018x+\frac{\pi }{4})$
where $x$ and $y$ are in $\mathrm{cm}$ and $t$ in $\mathrm{s}$. The positive direction of $x$ is from left to right.

What is the least distance between two successive crests in the wave?

(i) Transverse wave is a wave motion in which the particles of the medium vibrate about their mean positions at right angle to the direction of the propagation of the wave.

(ii) Longitudinal wave is a wave motion in which the particles of the medium vibrate about their mean positions along the direction of the propagation of the wave

2. Wavelength:

(i) Wavelength is the distance between two successive particles of the medium which are in phase.

(ii) The relation between wave-velocity, frequency and wavelength is given by $v= f\lambda$

3. Speed of transverse wave in a stretched string:

Speed of a transverse wave in a stretched string is given by $v= \sqrt{\frac{\mathrm{T}}{\mu }}$

4. Speed of sound: 

(i) Speed of sound in air is $v= \sqrt{\frac{\gamma \mathrm{P}}{\rho }}$

(ii) Velocity of sound in gases varies as directly proportional to square root of the temperature of the gas. $v\propto \sqrt{T}$

(iii) When temperature of the medium changes by 1 K or 1 °C, velocity of sound changes by 61 cm s-1.

(iv) Velocity of sound increases with humidity. Increase in humidity means decrease in the density of air

(v) Velocity of sound is independent of pressure of the gas.

5. Audible Range of Sound:

(i) Sound waves having frequency between $20 \mathrm{Hz}$ to $20,000 \mathrm{Hz}$ is audible to human ear. This frequency range is called audible frequency range.

(ii) Sound waves of frequency less than $20 \mathrm{Hz}$ are called infrasonic waves.

(iii) Sound waves of frequency more than $20,000 \mathrm{Hz}$ are known as ultrasonic waves.

6. Equation of a Travelling Wave:

A travelling wave or a progressive wave propagating along x-axis is represented by $y= a\mathrm{ sin}\left(kx\pm \omega t\right)$
Where $k= \frac{2\pi }{\lambda }$ is propagation constant or angular wave number, and $\omega = \frac{2\pi }{\mathrm{T}}$ is angular frequency.

7. Interference:

(i) Phase difference (Δϕ) and path difference (Δx) are related to each other as: $\mathrm{\Delta }\varphi = \frac{2\pi }{\lambda }\mathrm{\Delta }x$

(ii) Conditions for Constructive interference is phase difference between two interfering waves must be even multiple of π i.e.. $∆\varphi \mathrm{ }= 2n\pi$ where $n= 0,\mathrm{ }1,\mathrm{ }2,\mathrm{ }3,\dots .$

(iii) Conditions for Destructive interference is phase difference between two interfering waves must be odd multiple of π. i.e., $∆\varphi = \left(2n+1\right)\pi$ where $n= 0, 1, 2, 3,\dots .$

8. Reflection of Waves:

A phase difference of π exists between the incident and reflected waves if reflection takes place at a rigid boundary and no phase difference exists between the incident and reflected waves if reflection takes place at the free or open boundary.

9. Standing wave:

(i) Standing wave formed due to two travelling waves $y_1= A\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{ }(kx-\omega t)$ and $y_2= A\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}(kx+\omega t)$. It can be represented as $y= \left(2a\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{ }kx\right)\cos \omega t.$

(ii) The distance between any two successive nodes or anti-nodes is λ2.

(iii) The frequency of vibration of string fixed at both the ends is given by $f= \frac{nv}{2\mathrm{L}}= \frac{n}{2\mathrm{L}}\sqrt{\frac{\mathrm{T}}{\mu }}$
Frequency when $n= 1$, is called fundamental frequency or first harmonic
Frequency when $n= 2$, i.e., $f= 2f_0$ is called second harmonic and so on

(iv) The frequency of vibration of air column in a pipe closed at one end is $f= \left(n+\frac{1}{2}\right)\frac{v}{2\mathrm{L}}$
Frequency when n=0 is called fundamental frequency or first harmonic
Frequency when n=1, i.e., $f= 3f_0$ is called third harmonic and so on

10. Beats:

(i) The phenomenon of regular rise and fall in the intensity of sound, when two waves of nearly equal frequencies travelling along the same line and in the same direction superimpose each other is called beats.

(ii) Beat frequency, $f= f_1-f_2$.

11. Doppler’s effect in sound:

The apparent frequency heard by the observer $f= \left(\frac{v\pm v_0}{v\pm v_s}\right)f_0$
Here,
v is the speed of sound in the medium
vo is the velocity of the observer
vs is the velocity of the source
f0 is the frequency of the source