Question

1. express in words and mathematically the relationship between

a. period and frequency

b. wavelength and frequency

c. wavelength and period

Explanation:

Part a: Period and Frequency

• In words: Period ( \( T \)) is the time taken for one complete cycle of a wave to occur, and frequency ( \( f \)) is the number of cycles per unit time. They are reciprocal to each other, meaning as one increases, the other decreases.
• Mathematically: \( T=\frac{1}{f} \) or \( f = \frac{1}{T} \)

• In words: Wavelength ( \( \lambda \)) is the distance between two consecutive points in phase on a wave, and frequency ( \( f \)) is the number of cycles per unit time. For a wave traveling at a constant speed (usually the speed of light \( c \) for electromagnetic waves), wavelength and frequency are inversely proportional. As frequency increases, wavelength decreases, and vice versa.
• Mathematically: For a wave with speed \( v \) (e.g., \( v = c \) for light in vacuum), \( v=\lambda f \), so \( \lambda=\frac{v}{f} \) or \( f=\frac{v}{ \lambda} \)

• In words: Wavelength ( \( \lambda \)) is the distance between two consecutive points in phase on a wave, and period ( \( T \)) is the time taken for one complete cycle. For a wave traveling at speed \( v \), wavelength is equal to the product of the wave speed and the period. As the period increases (longer time for one cycle), the wavelength also increases if the speed is constant.
• Mathematically: Since \( v=\lambda f \) and \( f=\frac{1}{T} \), we can substitute \( f \) to get \( v=\frac{\lambda}{T} \), so \( \lambda = vT \)

Answer:

• In words: Period is the reciprocal of frequency (or frequency is the reciprocal of period).
• Mathematically: \( T=\frac{1}{f} \) (or \( f=\frac{1}{T} \))

Part b: Wavelength and Frequency