# Phase shift vs. horizontal shift, and frequency vs. angular frequency in sinusoidal functions

TL;DR version:

It seems to me that high school curricula no longer distinguish between "horizontal shift" and "phase shift", or between "frequency" and "angular frequency", when teaching sinusoidal functions. Is this perception accurate, and if so is this a recent change in usage?

Long version:

When not at my day job, I tutor a few high school students, and I recently I have noticed that some of them seem to be learning to describe sinusoidal functions in a way that differs from how I understand important words. In particular, the distinction between "horizontal shift" and "phase shift" seems to be elided, as is the distinction between "frequency" and "angular frequency".

Let me use an example to illustrate what I mean.

$$H(t) = 8 \sin \left( \frac{2\pi}{5}(t - 2) \right) + 9$$ which can be written in the equivalent form $$H(t) = 8 \sin \left( \frac{2\pi}{5}t - \frac{4\pi}5 \right) + 9$$

Here let's assume that $$t$$ is time, measured in seconds, and $$H(t)$$ is the height of something, measured in meters. My understanding is that the correct way to describe the features of this function is:

• The amplitude is $$8$$ meters.
• The midline is $$y = 9$$ meters (this describes the equilibrium position of the oscillating object).
• The period is $$5$$ seconds.
• The frequency is $$\frac 15$$ cycles/second, i.e. $$\frac 15$$ Hz.
• The angular frequency is $$\frac{2\pi}5$$ radians/second; note that this is the same as the coefficient of $$t$$ inside the argument of the sine function.
• The horizontal shift is $$t = 2$$ seconds; this means that the graph has been shifted $$2$$ seconds to the right, and therefore, when graphing the first'' period of the wave, we would begin at $$t = 2$$, rather than at $$t = 0$$.
• The phase shift is $$t = \frac{4\pi}5$$ radians; the interpretation of this is that the graph has been shifted $$\frac 25$$ of a cycle to the right (because a full cycle is $$2\pi$$ radians).

What I am seeing:

• Students are being taught that in an expression of the form $$A \sin (bt)$$, the constant $$b$$ is called the "frequency", and that the period is computed as $$T = \frac{2\pi}{b}$$. I agree with the latter formula, but not the word attached to the constant $$b$$. Frequency is measured in Hz, so it must be the reciprocal of the period -- no?

• Students are also being taught that in an expression of the form $$A \sin (b(t-h))$$, the quantity $$h$$ is called both "horizontal shift" and "phase shift" -- apparently these are synonyms now? The quantity $$bh$$, which is what I would call the "phase shift", does not seem to have a name, and expressions like $$\sin(3t - \pi)$$ must first be rewritten as $$\sin\left( 3 \left(t - \frac{\pi}3 \right) \right)$$, so that the "phase shift" of $$\pi/3$$ can be identified.

I would chalk this up to careless teachers, but I'm finding this in published (online) materials as well. For example, here is a CK-12 textbook that explicitly says "Phase shift is the horizontal shift left or right for periodic functions", and here is the same book defining frequency as "the number of cycles that occur in $$2\pi$$".

Now I am not by any means a language prescriptivist; it's fine with me if the meaning of words changes, or if words are used in different ways by different communities. (All definitions are conventions, and all conventions are local.) But I am curious where the boundaries of these communities are, and in who uses words in specific ways.

So now, my questions:

• In what curricula are the distinctions "frequency"/"angular frequency" and "horizontal shift"/"phase shift" preserved, and in which are they elided? (Answers might start with: "I teach X in country Y, and the convention here is...")
• Have these usages changed over time -- and in particular, is my sense that this is a recent shift accurate?

To a lesser extent I am interested in discussion of the pros and cons of both approaches, but primarily I am looking not for opinions, but for factual information about how these words are used, and by whom.

• Interesting. I graduated hs in '74 and college (with BA in math) in '79. I started teaching trig at a university in '86. I don't recall ever seeing these distinctions. Apr 9, 2021 at 14:27

## 2 Answers

These books are simply reflecting the longstanding and universal usage in physics and engineering, which is that these words can have either meaning, and any ambiguity is normally either resolved by context or unimportant.

In engineering and physics, in the English literature, angular frequency is oftentimes abbreviated in frequency. Sometimes this is explicitly stated at the beginning, but mostly it is given implicitly. However, the intended audience is expected to not make any confusion. The usage is certainly not recent, and I think I've seen it already about 40 years ago in even older books.

It's worth noting that other languages employ instead two very distinct terms for the two concepts, and there's less risk of confusing abbreviations. For instance, in Italian and French the angular frequency is called pulsazione and pulsation, respectively. Similar terms are probably used also in Spanish and German. Actually, English has a similarly distinct term too, pulsatance, but this has never become widespread, and it'd probably understood by very few people. For a list of these terms in various languages see the entry at the The World's Online Electrotechnical Vocabulary.

For what concerns phase shift and horizontal shift, I've never found authors who would not distinguish between the two, even though what you call horizontal shift is more commonly called time shift, or similarly for other quantities. I'd say that your example can be an idiosyncrasy limited to few people.

Finally, in engineering, what you call midline is more commonly called DC component, DC offset or vertical offset.