Does resonance still occur at half, a quarter, etc. of the resonant frequency?

Examples of resonance that I have seen are pushing a swing and shattering a glass.

I know the swing analogy, that if you push at the right frequency, you can make the swing go higher and higher. My confusion stems from this reasoning that I have had:

• When pushing a swing, instead of pushing it every time, if you pushed it every 2nd time it came back, it still works to increase the amplitude of the swing, i.e. creating resonance.
• So basically what this means is that by pushing the swing with half the resonant frequency still creates resonance.
• Then applying this to shattering a glass. If the glass had a resonant frequency of 880 Hz, it would mean that you need to sing the note A5 to shatter it. Using the conclusion from the swing example, this means that I should be able to sing the note an octave below, an A4 / 440 Hz note, and it would still work to be able to shatter the glass (although it probably would take longer as the glass is only having energy added to it every 2nd oscillation).

So while it's always like, an opera singer singing a really high note to shatter a glass, shouldn't it still work if you sang a note an octave below (because maybe you can't reach high enough for the real resonant frequency), or even two octaves (which would be equivalent to adding energy every 4th oscillation)?

However, since I've never heard of this, I get the feeling that I'm probably not understanding something and have some fallacies in my reasoning.

• Non-expert hence a comment rather than an answer. In the swing example, you are boosting the swing every cycle but doing nothing on the others so a net boost. In the opera singer case, the $440$Hz note will have an opposite effect on the $880$Hz résonance every second cycle. Commented Jul 21, 2022 at 14:11
• Thanks for accepting my answer. I'm sorry that you didn't get other answers too; it's good to have choice. I hope I made the point clearly that the swing and the wine glass are different sorts of system. The swing has only one natural frequency; the wine glass, many – which are not simply related to each other (unlike those for a violin string). Commented Jul 22, 2022 at 15:54
• Thanks for your answer. I'm currently only learning basic physics in high school, but the main things I realised was how the pulse is different to a periodic driving force, and at submultiple of the resonant frequency there would be parts which are pushing the wrong way. And for the wine glass, that its just more complicated. Commented Jul 24, 2022 at 10:19