VIDEO
HIGHLIGHT
Log InSign up
Derivation of beat frequency formula | Physics | Khan Academy
SummaryTranscriptChat

Derivation of beat frequency formula | Physics | Khan Academy

Deriving the Formula for Beat Frequency

In this section, the teacher introduces the concept of beat frequency and explains how it is derived. The formula for beat frequency is discussed, as well as its relationship to overlapping waves with different frequencies.

Understanding Beat Frequency

  • Beat frequency is produced when two waves with different frequencies overlap.
  • The number of times the waves go from constructive to destructive and back per second determines the beat frequency.
  • The formula for beat frequency is derived by taking the difference between the individual frequencies of the overlapping waves.

Deriving the Formula for Beat Frequency

  • To derive the formula, we first focus on finding the beat period, which represents the time it takes for one complete cycle of constructive to constructive.
  • By finding the beat period, we can determine that beat frequency is equal to one divided by the period.
  • The goal is to find how long it takes for two overlapping waves to become constructive again.

Analyzing Wave Phases

  • To find this time, we examine how these waves shift in phase over time.
  • Assuming they start off in phase (constructive), we want to know how long it takes for them to become constructive again after shifting out of phase.
  • We observe that every time one wave completes a full cycle (period), it becomes slightly ahead or behind in phase compared to the other wave.

Calculating Time Shift

  • The amount of phase shift between two peaks is determined by the difference in periods between both waves.
  • By multiplying this difference by the number of cycles completed by one wave, we can calculate how far apart these peaks are spaced in time at any given moment.

Constructive Again Condition

  • When these peaks are spaced apart by exactly one entire period of the second wave, they become constructive again.
  • This condition indicates that enough time has passed for the waves to overlap peak to peak.

Solving for Time

  • Although there is no variable "t" in the equation, we can introduce it by considering the periods of both waves as constants.
  • By finding the time when the distance between peaks equals one entire period of the second wave, we determine how long it took for them to become constructive again.

Sneaking "t" into the Equation

In this section, the teacher explains how to incorporate time ("t") into the equation despite its absence. The concept of relative phase and using a reference point are discussed.

Incorporating Time into the Equation

  • Despite not having an explicit variable for time ("t"), we can still include it in our calculations.
  • We introduce "t" by considering a reference point and measuring the time it takes for two peaks to become constructive again.

Relative Phase and Reference Point

  • By choosing a reference point where both waves start off in phase (constructive), we can measure how far apart they are at any given moment.
  • This allows us to determine when they become constructive again by comparing their relative phases.

Finding Constructive Again Time

  • The time it takes for two peaks to become constructive again is determined by when their separation equals one entire period of the second wave.
  • By solving this equation, we can find "t," which represents how long it took for them to reach that state.

The transcript does not provide further content beyond this timestamp.

New Section

In this section, the speaker discusses the concept of cycles and introduces a more general way to represent the number of cycles a wave has gone through.

Representing the Number of Cycles

  • The number of cycles a wave has gone through can be determined by dividing the time waited by the period of the wave. This is denoted as nR = t / T1.
  • By using this representation, it becomes possible to calculate the number of cycles even when it is not an integer value.

New Section

In this section, the speaker explains how to calculate the time it takes for two waves to get out of phase by a whole period and introduces the concept of beat period.

Calculating Time for Waves to Get Out of Phase

  • The time it takes for two waves to get out of phase by a whole period is denoted as t.
  • It can be calculated using the formula: t = (T1 * T2) / (T1 - T2), where T1 is the period of one wave and T2 is the period of another wave.
  • This time represents the beat period between two waves.

New Section

In this section, the speaker derives the formula for beat frequency from the previously calculated beat period.

Deriving Beat Frequency Formula

  • The formula for beat frequency can be derived by taking one over the beat period.
  • One over beat period equals one over (T2 - T1) / (T1 * T2).
  • Simplifying further, we get: 1 / (beat period) = (frequency 2) - (frequency 1).
  • Therefore, beat frequency equals frequency 2 minus frequency 1.

New Section

In this section, the speaker concludes the discussion by confirming that the derived formula matches the desired beat frequency formula.

Confirming Beat Frequency Formula

  • The derived beat frequency formula matches the desired formula: beat frequency = |frequency 2 - frequency 1|.
  • The absolute values are not necessary if we ensure that the larger period is subtracted from the smaller period.
  • The derived formula accurately represents how long one must wait for two waves to get back into phase and explains why it takes exactly that amount of time.

The transcript provided does not contain any additional sections or timestamps.

Video description

In this video David derives the formula for beat frequency. Created by David SantoPietro. Watch the next lesson: https://www.khanacademy.org/science/physics/fluids/density-and-pressure/v/specific-gravity?utm_source=YT&utm_medium=Desc&utm_campaign=physics Missed the previous lesson? https://www.khanacademy.org/science/physics/mechanical-waves-and-sound/standing-waves/v/beat-frequency?utm_source=YT&utm_medium=Desc&utm_campaign=physics Physics on Khan Academy: Physics is the study of the basic principles that govern the physical world around us. We'll start by looking at motion itself. Then, we'll learn about forces, momentum, energy, and other concepts in lots of different physical situations. To get the most out of physics, you'll need a solid understanding of algebra and a basic understanding of trigonometry. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. For free. For everyone. Forever. #YouCanLearnAnything Subscribe to Khan Academy’s Physics channel: https://www.youtube.com/channel/UC0oGarQW2lE5PxhGoQAKV7Q?sub_confirmation=1 Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy